stl_tree.h

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// RB tree implementation -*- C++ -*-
 
// Copyright (C) 2001-2026 Free Software Foundation, Inc.
//
// This file is part of the GNU ISO C++ Library.  This library is free
// software; you can redistribute it and/or modify it under the
// terms of the GNU General Public License as published by the
// Free Software Foundation; either version 3, or (at your option)
// any later version.
 
// This library is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
// GNU General Public License for more details.
 
// Under Section 7 of GPL version 3, you are granted additional
// permissions described in the GCC Runtime Library Exception, version
// 3.1, as published by the Free Software Foundation.
 
// You should have received a copy of the GNU General Public License and
// a copy of the GCC Runtime Library Exception along with this program;
// see the files COPYING3 and COPYING.RUNTIME respectively.  If not, see
// <http://www.gnu.org/licenses/>.
 
/*
 *
 * Copyright (c) 1996,1997
 * Silicon Graphics Computer Systems, Inc.
 *
 * Permission to use, copy, modify, distribute and sell this software
 * and its documentation for any purpose is hereby granted without fee,
 * provided that the above copyright notice appear in all copies and
 * that both that copyright notice and this permission notice appear
 * in supporting documentation.  Silicon Graphics makes no
 * representations about the suitability of this software for any
 * purpose.  It is provided "as is" without express or implied warranty.
 *
 *
 * Copyright (c) 1994
 * Hewlett-Packard Company
 *
 * Permission to use, copy, modify, distribute and sell this software
 * and its documentation for any purpose is hereby granted without fee,
 * provided that the above copyright notice appear in all copies and
 * that both that copyright notice and this permission notice appear
 * in supporting documentation.  Hewlett-Packard Company makes no
 * representations about the suitability of this software for any
 * purpose.  It is provided "as is" without express or implied warranty.
 *
 *
 */
 
/** @file bits/stl_tree.h
 *  This is an internal header file, included by other library headers.
 *  Do not attempt to use it directly. @headername{map,set}
 */
 
#ifndef _STL_TREE_H
#define _STL_TREE_H 1
 
#ifdef _GLIBCXX_SYSHDR
#pragma GCC system_header
#endif
 
#include <bits/stl_algobase.h>
#include <bits/allocator.h>
#include <bits/stl_function.h>
#include <bits/cpp_type_traits.h>
#include <bits/ptr_traits.h>
#include <ext/alloc_traits.h>
#if __cplusplus >= 201103L
# include <ext/aligned_buffer.h>
#endif
#ifdef __glibcxx_node_extract // >= C++17
# include <bits/node_handle.h>
#endif
 
#if __cplusplus < 201103L
# undef _GLIBCXX_USE_ALLOC_PTR_FOR_RB_TREE
# define _GLIBCXX_USE_ALLOC_PTR_FOR_RB_TREE 0
#elif ! defined _GLIBCXX_USE_ALLOC_PTR_FOR_RB_TREE
# define _GLIBCXX_USE_ALLOC_PTR_FOR_RB_TREE 1
#endif
 
namespace std _GLIBCXX_VISIBILITY(default)
{
_GLIBCXX_BEGIN_NAMESPACE_VERSION
 
  // Red-black tree class, designed for use in implementing STL
  // associative containers (set, multiset, map, and multimap). The
  // insertion and deletion algorithms are based on those in Cormen,
  // Leiserson, and Rivest, Introduction to Algorithms (MIT Press,
  // 1990), except that
  //
  // (1) the header cell is maintained with links not only to the root
  // but also to the leftmost node of the tree, to enable constant
  // time begin(), and to the rightmost node of the tree, to enable
  // linear time performance when used with the generic set algorithms
  // (set_union, etc.)
  //
  // (2) when a node being deleted has two children its successor node
  // is relinked into its place, rather than copied, so that the only
  // iterators invalidated are those referring to the deleted node.
 
  enum _Rb_tree_color { _S_red = false, _S_black = true };
 
  struct _Rb_tree_node_base
  {
    typedef _Rb_tree_node_base* _Base_ptr;
 
    _Rb_tree_color      _M_color;
    _Base_ptr           _M_parent;
    _Base_ptr           _M_left;
    _Base_ptr           _M_right;
 
    static _Base_ptr
    _S_minimum(_Base_ptr __x) _GLIBCXX_NOEXCEPT
    {
      while (__x->_M_left != 0) __x = __x->_M_left;
      return __x;
    }
 
    static _Base_ptr
    _S_maximum(_Base_ptr __x) _GLIBCXX_NOEXCEPT
    {
      while (__x->_M_right != 0) __x = __x->_M_right;
      return __x;
    }
 
    // This is not const-correct, but it's only used in a const access path
    // by std::_Rb_tree::_M_end() where the pointer is used to initialize a
    // const_iterator and so constness is restored.
    _Base_ptr
    _M_base_ptr() const _GLIBCXX_NOEXCEPT
    { return const_cast<_Rb_tree_node_base*>(this); }
  };
 
  // Helper type offering value initialization guarantee on the compare functor.
  template<typename _Key_compare>
    struct _Rb_tree_key_compare
    {
      _Key_compare              _M_key_compare;
 
      _Rb_tree_key_compare()
      _GLIBCXX_NOEXCEPT_IF(
        is_nothrow_default_constructible<_Key_compare>::value)
      : _M_key_compare()
      { }
 
      _Rb_tree_key_compare(const _Key_compare& __comp)
      : _M_key_compare(__comp)
      { }
 
#if __cplusplus >= 201103L
      // Copy constructor added for consistency with C++98 mode.
      _Rb_tree_key_compare(const _Rb_tree_key_compare&) = default;
 
      _Rb_tree_key_compare(_Rb_tree_key_compare&& __x)
        noexcept(is_nothrow_copy_constructible<_Key_compare>::value)
      : _M_key_compare(__x._M_key_compare)
      { }
#endif
    };
 
  // Helper type to manage default initialization of node count and header.
  struct _Rb_tree_header
  {
    _Rb_tree_node_base  _M_header;
    size_t              _M_node_count; // Keeps track of size of tree.
 
    _Rb_tree_header() _GLIBCXX_NOEXCEPT
    {
      _M_header._M_color = _S_red;
      _M_reset();
    }
 
#if __cplusplus >= 201103L
    _Rb_tree_header(_Rb_tree_header&& __x) noexcept
    {
      if (__x._M_header._M_parent != nullptr)
        _M_move_data(__x);
      else
        {
          _M_header._M_color = _S_red;
          _M_reset();
        }
    }
#endif
 
    void
    _M_move_data(_Rb_tree_header& __from)
    {
      _M_header._M_color = __from._M_header._M_color;
      _M_header._M_parent = __from._M_header._M_parent;
      _M_header._M_left = __from._M_header._M_left;
      _M_header._M_right = __from._M_header._M_right;
      _M_header._M_parent->_M_parent = &_M_header;
      _M_node_count = __from._M_node_count;
 
      __from._M_reset();
    }
 
    void
    _M_reset()
    {
      _M_header._M_parent = 0;
      _M_header._M_left = &_M_header;
      _M_header._M_right = &_M_header;
      _M_node_count = 0;
    }
  };
 
  template<typename _Val>
    struct _Rb_tree_node : public _Rb_tree_node_base
    {
#if __cplusplus < 201103L
      _Val _M_value_field;
 
      _Val*
      _M_valptr()
      { return std::__addressof(_M_value_field); }
 
      const _Val*
      _M_valptr() const
      { return std::__addressof(_M_value_field); }
#else
      __gnu_cxx::__aligned_membuf<_Val> _M_storage;
 
      _Val*
      _M_valptr()
      { return _M_storage._M_ptr(); }
 
      const _Val*
      _M_valptr() const
      { return _M_storage._M_ptr(); }
#endif
 
      _Rb_tree_node*
      _M_node_ptr() _GLIBCXX_NOEXCEPT
      { return this; }
    };
 
#if _GLIBCXX_USE_ALLOC_PTR_FOR_RB_TREE
namespace __rb_tree
{
  template<typename _VoidPtr>
    struct _Node_base
    {
      using _Base_ptr = __ptr_rebind<_VoidPtr, _Node_base>;
 
      _Rb_tree_color    _M_color;
      _Base_ptr         _M_parent;
      _Base_ptr         _M_left;
      _Base_ptr         _M_right;
 
      static _Base_ptr
      _S_minimum(_Base_ptr __x) _GLIBCXX_NOEXCEPT
      {
        while (__x->_M_left) __x = __x->_M_left;
        return __x;
      }
 
      static _Base_ptr
      _S_maximum(_Base_ptr __x) _GLIBCXX_NOEXCEPT
      {
        while (__x->_M_right) __x = __x->_M_right;
        return __x;
      }
 
      // This is not const-correct, but it's only used in a const access path
      // by std::_Rb_tree::_M_end() where the pointer is used to initialize a
      // const_iterator and so constness is restored.
      _Base_ptr
      _M_base_ptr() const noexcept
      {
        return pointer_traits<_Base_ptr>::pointer_to
          (*const_cast<_Node_base*>(this));
      }
    };
 
  // Helper type to manage default initialization of node count and header.
  template<typename _NodeBase>
    struct _Header
    {
    private:
      using _Base_ptr =  typename _NodeBase::_Base_ptr;
 
    public:
      _NodeBase         _M_header;
      size_t            _M_node_count; // Keeps track of size of tree.
 
      _Header() noexcept
      {
        _M_header._M_color = _S_red;
        _M_reset();
      }
 
      _Header(_Header&& __x) noexcept
      {
        if (__x._M_header._M_parent)
          _M_move_data(__x);
        else
          {
            _M_header._M_color = _S_red;
            _M_reset();
          }
      }
 
      void
      _M_move_data(_Header& __from)
      {
        _M_header._M_color = __from._M_header._M_color;
        _M_header._M_parent = __from._M_header._M_parent;
        _M_header._M_left = __from._M_header._M_left;
        _M_header._M_right = __from._M_header._M_right;
        _M_header._M_parent->_M_parent = _M_header._M_base_ptr();
        _M_node_count = __from._M_node_count;
 
        __from._M_reset();
      }
 
      void
      _M_reset()
      {
        _M_header._M_parent = nullptr;
        _M_header._M_left = _M_header._M_right = _M_header._M_base_ptr();
        _M_node_count = 0;
      }
    };
 
  template<typename _ValPtr>
    struct _Node : public __rb_tree::_Node_base<__ptr_rebind<_ValPtr, void>>
    {
      using value_type = typename pointer_traits<_ValPtr>::element_type;
      using _Node_ptr = __ptr_rebind<_ValPtr, _Node>;
 
      _Node() noexcept { }
      ~_Node() { }
      _Node(_Node&&) = delete;
 
      union _Uninit_storage
      {
        _Uninit_storage() noexcept { }
        ~_Uninit_storage() { }
 
        value_type _M_data;
      };
      _Uninit_storage _M_u;
 
      value_type*
      _M_valptr()
      { return std::addressof(_M_u._M_data); }
 
      value_type const*
      _M_valptr() const
      { return std::addressof(_M_u._M_data); }
 
      _Node_ptr
      _M_node_ptr() noexcept
      { return pointer_traits<_Node_ptr>::pointer_to(*this); }
    };
} // namespace __rb_tree
#endif // _GLIBCXX_USE_ALLOC_PTR_FOR_RB_TREE
 
  _GLIBCXX_PURE _Rb_tree_node_base*
  _Rb_tree_increment(_Rb_tree_node_base* __x) throw ();
 
  _GLIBCXX_PURE _Rb_tree_node_base*
  _Rb_tree_decrement(_Rb_tree_node_base* __x) throw ();
 
  template<typename _Tp>
    struct _Rb_tree_iterator
    {
      typedef _Tp  value_type;
      typedef _Tp& reference;
      typedef _Tp* pointer;
 
      typedef bidirectional_iterator_tag iterator_category;
      typedef ptrdiff_t                  difference_type;
 
      typedef _Rb_tree_node_base::_Base_ptr     _Base_ptr;
      typedef _Rb_tree_node<_Tp>*               _Node_ptr;
 
      _Rb_tree_iterator() _GLIBCXX_NOEXCEPT
      : _M_node() { }
 
      explicit
      _Rb_tree_iterator(_Base_ptr __x) _GLIBCXX_NOEXCEPT
      : _M_node(__x) { }
 
      reference
      operator*() const _GLIBCXX_NOEXCEPT
      { return *static_cast<_Node_ptr>(_M_node)->_M_valptr(); }
 
      pointer
      operator->() const _GLIBCXX_NOEXCEPT
      { return static_cast<_Node_ptr>(_M_node)->_M_valptr(); }
 
      _Rb_tree_iterator&
      operator++() _GLIBCXX_NOEXCEPT
      {
        _M_node = _Rb_tree_increment(_M_node);
        return *this;
      }
 
      _Rb_tree_iterator
      operator++(int) _GLIBCXX_NOEXCEPT
      {
        _Rb_tree_iterator __tmp = *this;
        _M_node = _Rb_tree_increment(_M_node);
        return __tmp;
      }
 
      _Rb_tree_iterator&
      operator--() _GLIBCXX_NOEXCEPT
      {
        _M_node = _Rb_tree_decrement(_M_node);
        return *this;
      }
 
      _Rb_tree_iterator
      operator--(int) _GLIBCXX_NOEXCEPT
      {
        _Rb_tree_iterator __tmp = *this;
        _M_node = _Rb_tree_decrement(_M_node);
        return __tmp;
      }
 
      friend bool
      operator==(const _Rb_tree_iterator& __x,
                 const _Rb_tree_iterator& __y) _GLIBCXX_NOEXCEPT
      { return __x._M_node == __y._M_node; }
 
#if ! __cpp_lib_three_way_comparison
      friend bool
      operator!=(const _Rb_tree_iterator& __x,
                 const _Rb_tree_iterator& __y) _GLIBCXX_NOEXCEPT
      { return __x._M_node != __y._M_node; }
#endif
 
      _Base_ptr _M_node;
    };
 
  template<typename _Tp>
    struct _Rb_tree_const_iterator
    {
      typedef _Tp        value_type;
      typedef const _Tp& reference;
      typedef const _Tp* pointer;
 
      typedef _Rb_tree_iterator<_Tp> iterator;
 
      typedef bidirectional_iterator_tag iterator_category;
      typedef ptrdiff_t                  difference_type;
 
      typedef _Rb_tree_node_base::_Base_ptr     _Base_ptr;
      typedef const _Rb_tree_node<_Tp>*         _Node_ptr;
 
      _Rb_tree_const_iterator() _GLIBCXX_NOEXCEPT
      : _M_node() { }
 
      explicit
      _Rb_tree_const_iterator(_Base_ptr __x) _GLIBCXX_NOEXCEPT
      : _M_node(__x) { }
 
      _Rb_tree_const_iterator(const iterator& __it) _GLIBCXX_NOEXCEPT
      : _M_node(__it._M_node) { }
 
      reference
      operator*() const _GLIBCXX_NOEXCEPT
      { return *static_cast<_Node_ptr>(_M_node)->_M_valptr(); }
 
      pointer
      operator->() const _GLIBCXX_NOEXCEPT
      { return static_cast<_Node_ptr>(_M_node)->_M_valptr(); }
 
      _Rb_tree_const_iterator&
      operator++() _GLIBCXX_NOEXCEPT
      {
        _M_node = _Rb_tree_increment(_M_node);
        return *this;
      }
 
      _Rb_tree_const_iterator
      operator++(int) _GLIBCXX_NOEXCEPT
      {
        _Rb_tree_const_iterator __tmp = *this;
        _M_node = _Rb_tree_increment(_M_node);
        return __tmp;
      }
 
      _Rb_tree_const_iterator&
      operator--() _GLIBCXX_NOEXCEPT
      {
        _M_node = _Rb_tree_decrement(_M_node);
        return *this;
      }
 
      _Rb_tree_const_iterator
      operator--(int) _GLIBCXX_NOEXCEPT
      {
        _Rb_tree_const_iterator __tmp = *this;
        _M_node = _Rb_tree_decrement(_M_node);
        return __tmp;
      }
 
      friend bool
      operator==(const _Rb_tree_const_iterator& __x,
                 const _Rb_tree_const_iterator& __y) _GLIBCXX_NOEXCEPT
      { return __x._M_node == __y._M_node; }
 
#if ! __cpp_lib_three_way_comparison
      friend bool
      operator!=(const _Rb_tree_const_iterator& __x,
                 const _Rb_tree_const_iterator& __y) _GLIBCXX_NOEXCEPT
      { return __x._M_node != __y._M_node; }
#endif
 
      _Base_ptr _M_node;
    };
 
  __attribute__((__nonnull__))
  void
  _Rb_tree_insert_and_rebalance(const bool __insert_left,
                                _Rb_tree_node_base* __x,
                                _Rb_tree_node_base* __p,
                                _Rb_tree_node_base& __header) throw ();
 
  __attribute__((__nonnull__,__returns_nonnull__))
  _Rb_tree_node_base*
  _Rb_tree_rebalance_for_erase(_Rb_tree_node_base* const __z,
                               _Rb_tree_node_base& __header) throw ();
 
namespace __rb_tree
{
#if _GLIBCXX_USE_ALLOC_PTR_FOR_RB_TREE
  template<bool _Const, typename _ValPtr>
    struct _Iterator
    {
      template<typename _Tp>
        using __maybe_const = __conditional_t<_Const, const _Tp, _Tp>;
 
      using __ptr_traits =      pointer_traits<_ValPtr>;
      using value_type =        typename __ptr_traits::element_type;
      using reference =         __maybe_const<value_type>&;
      using pointer =           __maybe_const<value_type>*;
 
      using iterator_category = bidirectional_iterator_tag;
      using difference_type =   ptrdiff_t;
 
      using _Node = __rb_tree::_Node<_ValPtr>;
      using _Node_base = __rb_tree::_Node_base<__ptr_rebind<_ValPtr, void>>;
      using _Base_ptr =  typename _Node_base::_Base_ptr;
 
      _Iterator() noexcept
      : _M_node() { }
 
      constexpr explicit
      _Iterator(_Base_ptr __x) noexcept
      : _M_node(__x) { }
 
      _Iterator(const _Iterator&) = default;
      _Iterator& operator=(const _Iterator&) = default;
 
#ifdef __glibcxx_concepts
      constexpr
      _Iterator(const _Iterator<false, _ValPtr>& __it) requires _Const
#else
      template<bool _OtherConst,
               typename = __enable_if_t<_Const && !_OtherConst>>
        constexpr
        _Iterator(const _Iterator<_OtherConst, _ValPtr>& __it)
#endif
        : _M_node(__it._M_node) { }
 
      [[__nodiscard__]]
      reference
      operator*() const noexcept
      { return *static_cast<_Node&>(*_M_node)._M_valptr(); }
 
      [[__nodiscard__]]
      pointer
      operator->() const noexcept
      { return static_cast<_Node&>(*_M_node)._M_valptr(); }
 
      _GLIBCXX14_CONSTEXPR _Iterator&
      operator++() noexcept
      {
        if (_M_node->_M_right)
          {
            _M_node = _M_node->_M_right;
            while (_M_node->_M_left)
              _M_node = _M_node->_M_left;
          }
        else
          {
            _Base_ptr __y = _M_node->_M_parent;
            while (_M_node == __y->_M_right)
              {
                _M_node = __y;
                __y = __y->_M_parent;
              }
            if (_M_node->_M_right != __y)
              _M_node = __y;
          }
 
        return *this;
      }
 
      _GLIBCXX14_CONSTEXPR _Iterator
      operator++(int) noexcept
      {
        _Iterator __tmp(this->_M_node);
        ++*this;
        return __tmp;
      }
 
      _GLIBCXX14_CONSTEXPR _Iterator&
      operator--() noexcept
      {
        if (_M_node->_M_color == _S_red
            && _M_node->_M_parent->_M_parent == _M_node)
          _M_node = _M_node->_M_right;
        else if (_M_node->_M_left)
          {
            _Base_ptr __y = _M_node->_M_left;
            while (__y->_M_right)
              __y = __y->_M_right;
            _M_node = __y;
          }
        else
          {
            _Base_ptr __y = _M_node->_M_parent;
            while (_M_node == __y->_M_left)
              {
                _M_node = __y;
                __y = __y->_M_parent;
              }
            _M_node = __y;
          }
        return *this;
      }
 
      _GLIBCXX14_CONSTEXPR _Iterator
      operator--(int) noexcept
      {
        _Iterator __tmp(this->_M_node);
        --*this;
        return __tmp;
      }
 
      [[__nodiscard__]]
      friend bool
      operator==(const _Iterator& __x, const _Iterator& __y) _GLIBCXX_NOEXCEPT
      { return __x._M_node == __y._M_node; }
 
#if ! __cpp_lib_three_way_comparison
      [[__nodiscard__]]
      friend bool
      operator!=(const _Iterator& __x, const _Iterator& __y) _GLIBCXX_NOEXCEPT
      { return __x._M_node != __y._M_node; }
#endif
 
      _Base_ptr _M_node;
    };
#endif // USE_ALLOC_PTR_FOR_RB_TREE
 
  // Determine the node and iterator types used by std::_Rb_tree.
  template<typename _Val, typename _Ptr>
    struct _Node_traits;
 
#if _GLIBCXX_USE_ALLOC_PTR_FOR_RB_TREE <= 9000
  // Specialization for the simple case where the allocator's pointer type
  // is the same type as value_type*.
  // For ABI compatibility we can't change the types used for this case.
  template<typename _Val>
    struct _Node_traits<_Val, _Val*>
    {
      typedef _Rb_tree_node<_Val>               _Node;
      typedef _Node*                            _Node_ptr;
      typedef _Rb_tree_node_base                _Node_base;
      typedef _Node_base*                       _Base_ptr;
      typedef _Rb_tree_header                   _Header_t;
      typedef _Rb_tree_iterator<_Val>           _Iterator;
      typedef _Rb_tree_const_iterator<_Val>     _Const_iterator;
 
      __attribute__((__nonnull__))
      static void
      _S_insert_and_rebalance(const bool __insert_left,
                              _Node_base* __x, _Node_base* __p,
                              _Node_base& __header) _GLIBCXX_USE_NOEXCEPT
      {
        return _Rb_tree_insert_and_rebalance(__insert_left, __x, __p, __header);
      }
 
      __attribute__((__nonnull__,__returns_nonnull__))
      static _Node_base*
      _S_rebalance_for_erase(_Node_base* const __z,
                             _Node_base& __header) _GLIBCXX_USE_NOEXCEPT
      { return _Rb_tree_rebalance_for_erase(__z, __header); }
    };
#endif
 
#if ! _GLIBCXX_USE_ALLOC_PTR_FOR_RB_TREE
  // Always use the T* specialization.
  template<typename _Val, typename _Ptr>
    struct _Node_traits
    : _Node_traits<_Val, _Val*>
    { };
#else
  // Primary template used when the allocator uses fancy pointers.
  template<typename _Val, typename _ValPtr>
    struct _Node_traits
    {
      using _Node = __rb_tree::_Node<_ValPtr>;
      using _Node_ptr = __ptr_rebind<_ValPtr, _Node>;
      using _Node_base = __rb_tree::_Node_base<__ptr_rebind<_ValPtr, void>>;
      using _Base_ptr = __ptr_rebind<_ValPtr, _Node_base>;
      using _Header_t = __rb_tree::_Header<_Node_base>;
      using _Iterator = __rb_tree::_Iterator<false, _ValPtr>;
      using _Const_iterator = __rb_tree::_Iterator<true, _ValPtr>;
 
      static void
      _Rotate_left(_Base_ptr __x, _Base_ptr& __root)
      {
        const _Base_ptr __y = __x->_M_right;
 
        __x->_M_right = __y->_M_left;
        if (__y->_M_left)
          __y->_M_left->_M_parent = __x;
        __y->_M_parent = __x->_M_parent;
 
        if (__x == __root)
          __root = __y;
        else if (__x == __x->_M_parent->_M_left)
          __x->_M_parent->_M_left = __y;
        else
          __x->_M_parent->_M_right = __y;
        __y->_M_left = __x;
        __x->_M_parent = __y;
      }
 
      static void
      _Rotate_right(_Base_ptr __x, _Base_ptr& __root)
      {
        const _Base_ptr __y = __x->_M_left;
 
        __x->_M_left = __y->_M_right;
        if (__y->_M_right)
          __y->_M_right->_M_parent = __x;
        __y->_M_parent = __x->_M_parent;
 
        if (__x == __root)
          __root = __y;
        else if (__x == __x->_M_parent->_M_right)
          __x->_M_parent->_M_right = __y;
        else
          __x->_M_parent->_M_left = __y;
        __y->_M_right = __x;
        __x->_M_parent = __y;
      }
 
      static void
      _S_insert_and_rebalance(const bool __insert_left,
                              _Base_ptr __x, _Base_ptr __p,
                              _Node_base& __header)
      {
        _Base_ptr& __root = __header._M_parent;
 
        // Initialize fields in new node to insert.
        __x->_M_parent = __p;
        __x->_M_left = __x->_M_right = nullptr;
        __x->_M_color = _S_red;
 
        // Insert.
        // Make new node child of parent and maintain root, leftmost and
        // rightmost nodes.
        // N.B. First node is always inserted left.
        if (__insert_left)
          {
            __p->_M_left = __x; // also makes leftmost = __x when __p == &__header
 
            if (std::__to_address(__p) == std::addressof(__header))
              {
                __header._M_parent = __x;
                __header._M_right = __x;
              }
            else if (__p == __header._M_left)
              __header._M_left = __x; // maintain leftmost pointing to min node
          }
        else
          {
            __p->_M_right = __x;
 
            if (__p == __header._M_right)
              __header._M_right = __x; // maintain rightmost pointing to max node
          }
        // Rebalance.
        while (__x != __root
               && __x->_M_parent->_M_color == _S_red)
          {
            const _Base_ptr __xpp = __x->_M_parent->_M_parent;
 
            if (__x->_M_parent == __xpp->_M_left)
              {
                const _Base_ptr __y = __xpp->_M_right;
                if (__y && __y->_M_color == _S_red)
                  {
                    __x->_M_parent->_M_color = _S_black;
                    __y->_M_color = _S_black;
                    __xpp->_M_color = _S_red;
                    __x = __xpp;
                  }
                else
                  {
                    if (__x == __x->_M_parent->_M_right)
                      {
                        __x = __x->_M_parent;
                        _Rotate_left(__x, __root);
                      }
                    __x->_M_parent->_M_color = _S_black;
                    __xpp->_M_color = _S_red;
                    _Rotate_right(__xpp, __root);
                  }
              }
            else
              {
                const _Base_ptr __y = __xpp->_M_left;
                if (__y && __y->_M_color == _S_red)
                  {
                    __x->_M_parent->_M_color = _S_black;
                    __y->_M_color = _S_black;
                    __xpp->_M_color = _S_red;
                    __x = __xpp;
                  }
                else
                  {
                    if (__x == __x->_M_parent->_M_left)
                      {
                        __x = __x->_M_parent;
                        _Rotate_right(__x, __root);
                      }
                    __x->_M_parent->_M_color = _S_black;
                    __xpp->_M_color = _S_red;
                    _Rotate_left(__xpp, __root);
                  }
              }
          }
        __root->_M_color = _S_black;
      }
 
      static _Base_ptr
      _S_rebalance_for_erase(_Base_ptr __z, _Node_base& __header)
      {
        _Base_ptr& __root = __header._M_parent;
        _Base_ptr& __leftmost = __header._M_left;
        _Base_ptr& __rightmost = __header._M_right;
        _Base_ptr __y = __z;
        _Base_ptr __x{};
        _Base_ptr __x_parent{};
 
        if (!__y->_M_left)     // __z has at most one non-null child. y == z.
          __x = __y->_M_right;     // __x might be null.
        else
          if (!__y->_M_right)  // __z has exactly one non-null child. y == z.
            __x = __y->_M_left;    // __x is not null.
          else
            {
              // __z has two non-null children.  Set __y to
              __y = __y->_M_right;   //   __z's successor.  __x might be null.
              while (__y->_M_left)
                __y = __y->_M_left;
              __x = __y->_M_right;
            }
        if (__y != __z)
          {
            // relink y in place of z.  y is z's successor
            __z->_M_left->_M_parent = __y;
            __y->_M_left = __z->_M_left;
            if (__y != __z->_M_right)
              {
                __x_parent = __y->_M_parent;
                if (__x)
                  __x->_M_parent = __y->_M_parent;
                __y->_M_parent->_M_left = __x;   // __y must be a child of _M_left
                __y->_M_right = __z->_M_right;
                __z->_M_right->_M_parent = __y;
              }
            else
              __x_parent = __y;
            if (__root == __z)
              __root = __y;
            else if (__z->_M_parent->_M_left == __z)
              __z->_M_parent->_M_left = __y;
            else
              __z->_M_parent->_M_right = __y;
            __y->_M_parent = __z->_M_parent;
            std::swap(__y->_M_color, __z->_M_color);
            __y = __z;
            // __y now points to node to be actually deleted
          }
        else
          {                        // __y == __z
            __x_parent = __y->_M_parent;
            if (__x)
              __x->_M_parent = __y->_M_parent;
            if (__root == __z)
              __root = __x;
            else
              if (__z->_M_parent->_M_left == __z)
                __z->_M_parent->_M_left = __x;
              else
                __z->_M_parent->_M_right = __x;
            if (__leftmost == __z)
              {
                if (!__z->_M_right)        // __z->_M_left must be null also
                  __leftmost = __z->_M_parent;
                // makes __leftmost == _M_header if __z == __root
                else
                  __leftmost = _Node_base::_S_minimum(__x);
              }
            if (__rightmost == __z)
              {
                if (__z->_M_left == 0)         // __z->_M_right must be null also
                  __rightmost = __z->_M_parent;
                // makes __rightmost == _M_header if __z == __root
                else                      // __x == __z->_M_left
                  __rightmost = _Node_base::_S_maximum(__x);
              }
          }
        if (__y->_M_color != _S_red)
          {
            while (__x != __root && (__x == 0 || __x->_M_color == _S_black))
              if (__x == __x_parent->_M_left)
                {
                  _Base_ptr __w = __x_parent->_M_right;
                  if (__w->_M_color == _S_red)
                    {
                      __w->_M_color = _S_black;
                      __x_parent->_M_color = _S_red;
                      _Rotate_left(__x_parent, __root);
                      __w = __x_parent->_M_right;
                    }
                  if ((!__w->_M_left || __w->_M_left->_M_color == _S_black) &&
                      (!__w->_M_right || __w->_M_right->_M_color == _S_black))
                    {
                      __w->_M_color = _S_red;
                      __x = __x_parent;
                      __x_parent = __x_parent->_M_parent;
                    }
                  else
                    {
                      if (!__w->_M_right || __w->_M_right->_M_color == _S_black)
                        {
                          __w->_M_left->_M_color = _S_black;
                          __w->_M_color = _S_red;
                          _Rotate_right(__w, __root);
                          __w = __x_parent->_M_right;
                        }
                      __w->_M_color = __x_parent->_M_color;
                      __x_parent->_M_color = _S_black;
                      if (__w->_M_right)
                        __w->_M_right->_M_color = _S_black;
                      _Rotate_left(__x_parent, __root);
                      break;
                    }
                }
              else
                {
                  // same as above, with _M_right <-> _M_left.
                  _Base_ptr __w = __x_parent->_M_left;
                  if (__w->_M_color == _S_red)
                    {
                      __w->_M_color = _S_black;
                      __x_parent->_M_color = _S_red;
                      _Rotate_right(__x_parent, __root);
                      __w = __x_parent->_M_left;
                    }
                  if ((!__w->_M_right || __w->_M_right->_M_color == _S_black) &&
                      (!__w->_M_left || __w->_M_left->_M_color == _S_black))
                    {
                      __w->_M_color = _S_red;
                      __x = __x_parent;
                      __x_parent = __x_parent->_M_parent;
                    }
                  else
                    {
                      if (!__w->_M_left || __w->_M_left->_M_color == _S_black)
                        {
                          __w->_M_right->_M_color = _S_black;
                          __w->_M_color = _S_red;
                          _Rotate_left(__w, __root);
                          __w = __x_parent->_M_left;
                        }
                      __w->_M_color = __x_parent->_M_color;
                      __x_parent->_M_color = _S_black;
                      if (__w->_M_left)
                        __w->_M_left->_M_color = _S_black;
                      _Rotate_right(__x_parent, __root);
                      break;
                    }
                }
            if (__x)
              __x->_M_color = _S_black;
          }
 
        return __y;
      }
    };
#endif
} // namespace __rb_tree
 
#ifdef __glibcxx_node_extract // >= C++17
  template<typename _Tree1, typename _Cmp2>
    struct _Rb_tree_merge_helper { };
#endif
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc = allocator<_Val> >
    class _Rb_tree
    {
      typedef typename __gnu_cxx::__alloc_traits<_Alloc>::template
        rebind<_Val>::other _Val_alloc_type;
 
      typedef __gnu_cxx::__alloc_traits<_Val_alloc_type> _Val_alloc_traits;
      typedef typename _Val_alloc_traits::pointer _ValPtr;
      typedef __rb_tree::_Node_traits<_Val, _ValPtr> _Node_traits;
 
      typedef typename _Node_traits::_Node_base         _Node_base;
      typedef typename _Node_traits::_Node              _Node;
 
      typedef typename __gnu_cxx::__alloc_traits<_Alloc>::template
        rebind<_Node>::other _Node_allocator;
 
      typedef __gnu_cxx::__alloc_traits<_Node_allocator> _Node_alloc_traits;
 
    protected:
      typedef typename _Node_traits::_Base_ptr  _Base_ptr;
      typedef typename _Node_traits::_Node_ptr  _Node_ptr;
 
    private:
      // Functor recycling a pool of nodes and using allocation once the pool
      // is empty.
      struct _Reuse_or_alloc_node
      {
        _Reuse_or_alloc_node(_Rb_tree& __t)
        : _M_root(__t._M_root()), _M_nodes(__t._M_rightmost()), _M_t(__t)
        {
          if (_M_root)
            {
              _M_root->_M_parent = _Base_ptr();
 
              if (_M_nodes->_M_left)
                _M_nodes = _M_nodes->_M_left;
            }
          else
            _M_nodes = _Base_ptr();
        }
 
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wc++11-extensions"
        _Reuse_or_alloc_node(const _Reuse_or_alloc_node&) = delete;
#pragma GCC diagnostic pop
 
        ~_Reuse_or_alloc_node()
        {
          if (_M_root)
            _M_t._M_erase(static_cast<_Node&>(*_M_root)._M_node_ptr());
        }
 
        template<typename _Arg>
          _Node_ptr
          operator()(_GLIBCXX_FWDREF(_Arg) __arg)
          {
            _Base_ptr __base = _M_extract();
            if (__base)
              {
                _Node_ptr __node = static_cast<_Node&>(*__base)._M_node_ptr();
                _M_t._M_destroy_node(__node);
                _M_t._M_construct_node(__node, _GLIBCXX_FORWARD(_Arg, __arg));
                return __node;
              }
 
            return _M_t._M_create_node(_GLIBCXX_FORWARD(_Arg, __arg));
          }
 
      private:
        _Base_ptr
        _M_extract()
        {
          if (!_M_nodes)
            return _M_nodes;
 
          _Base_ptr __node = _M_nodes;
          _M_nodes = _M_nodes->_M_parent;
          if (_M_nodes)
            {
              if (_M_nodes->_M_right == __node)
                {
                  _M_nodes->_M_right = _Base_ptr();
 
                  if (_M_nodes->_M_left)
                    {
                      _M_nodes = _M_nodes->_M_left;
 
                      while (_M_nodes->_M_right)
                        _M_nodes = _M_nodes->_M_right;
 
                      if (_M_nodes->_M_left)
                        _M_nodes = _M_nodes->_M_left;
                    }
                }
              else // __node is on the left.
                _M_nodes->_M_left = _Base_ptr();
            }
          else
            _M_root = _Base_ptr();
 
          return __node;
        }
 
        _Base_ptr _M_root;
        _Base_ptr _M_nodes;
        _Rb_tree& _M_t;
      };
 
      // Functor similar to the previous one but without any pool of nodes to
      // recycle.
      struct _Alloc_node
      {
        _Alloc_node(_Rb_tree& __t)
        : _M_t(__t) { }
 
        template<typename _Arg>
          _Node_ptr
          operator()(_GLIBCXX_FWDREF(_Arg) __arg) const
          { return _M_t._M_create_node(_GLIBCXX_FORWARD(_Arg, __arg)); }
 
      private:
        _Rb_tree& _M_t;
      };
 
    public:
      typedef _Key                              key_type;
      typedef _Val                              value_type;
      typedef value_type*                       pointer;
      typedef const value_type*                 const_pointer;
      typedef value_type&                       reference;
      typedef const value_type&                 const_reference;
      typedef size_t                            size_type;
      typedef ptrdiff_t                         difference_type;
      typedef _Alloc                            allocator_type;
 
      _Node_allocator&
      _M_get_Node_allocator() _GLIBCXX_NOEXCEPT
      { return this->_M_impl; }
 
      const _Node_allocator&
      _M_get_Node_allocator() const _GLIBCXX_NOEXCEPT
      { return this->_M_impl; }
 
      allocator_type
      get_allocator() const _GLIBCXX_NOEXCEPT
      { return allocator_type(_M_get_Node_allocator()); }
 
    protected:
      _Node_ptr
      _M_get_node()
      {
#if __cplusplus < 201102L || _GLIBCXX_USE_ALLOC_PTR_FOR_RB_TREE
        return _Node_alloc_traits::allocate(_M_get_Node_allocator(), 1);
#else
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wc++17-extensions" // if constexpr
        using __alloc_pointer = typename _Node_alloc_traits::pointer;
        if constexpr (is_same<_Node_ptr, __alloc_pointer>::value)
          return _Node_alloc_traits::allocate(_M_get_Node_allocator(), 1);
        else
          {
            auto __ptr =
              _Node_alloc_traits::allocate(_M_get_Node_allocator(), 1);
            return std::__to_address(__ptr);
          }
#pragma GCC diagnostic pop
#endif
      }
 
      void
      _M_put_node(_Node_ptr __p) _GLIBCXX_NOEXCEPT
      {
#if __cplusplus < 201102L || _GLIBCXX_USE_ALLOC_PTR_FOR_RB_TREE
        _Node_alloc_traits::deallocate(_M_get_Node_allocator(), __p, 1);
#else
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wc++17-extensions" // if constexpr
        using __alloc_pointer = typename _Node_alloc_traits::pointer;
        if constexpr (is_same<_Node_ptr, __alloc_pointer>::value)
          _Node_alloc_traits::deallocate(_M_get_Node_allocator(), __p, 1);
        else
          {
            // When not using the allocator's pointer type internally we must
            // convert __p to __alloc_pointer so it can be deallocated.
            auto __ap = pointer_traits<__alloc_pointer>::pointer_to(*__p);
            _Node_alloc_traits::deallocate(_M_get_Node_allocator(), __ap, 1);
          }
#pragma GCC diagnostic pop
#endif
      }
 
#if __cplusplus < 201103L
      void
      _M_construct_node(_Node_ptr __node, const value_type& __x)
      {
        __try
          { get_allocator().construct(__node->_M_valptr(), __x); }
        __catch(...)
          {
            _M_put_node(__node);
            __throw_exception_again;
          }
      }
 
      _Node_ptr
      _M_create_node(const value_type& __x)
      {
        _Node_ptr __tmp = _M_get_node();
        _M_construct_node(__tmp, __x);
        return __tmp;
      }
#else
      template<typename... _Args>
        void
        _M_construct_node(_Node_ptr __node, _Args&&... __args)
        {
          __try
            {
              ::new(std::addressof(*__node)) _Node;
              _Node_alloc_traits::construct(_M_get_Node_allocator(),
                                            __node->_M_valptr(),
                                            std::forward<_Args>(__args)...);
            }
          __catch(...)
            {
              __node->~_Node();
              _M_put_node(__node);
              __throw_exception_again;
            }
        }
 
      template<typename... _Args>
        _Node_ptr
        _M_create_node(_Args&&... __args)
        {
          _Node_ptr __tmp = _M_get_node();
          _M_construct_node(__tmp, std::forward<_Args>(__args)...);
          return __tmp;
        }
#endif
 
      void
      _M_destroy_node(_Node_ptr __p) _GLIBCXX_NOEXCEPT
      {
#if __cplusplus < 201103L
        get_allocator().destroy(__p->_M_valptr());
#else
        _Node_alloc_traits::destroy(_M_get_Node_allocator(), __p->_M_valptr());
        __p->~_Node();
#endif
      }
 
      void
      _M_drop_node(_Node_ptr __p) _GLIBCXX_NOEXCEPT
      {
        _M_destroy_node(__p);
        _M_put_node(__p);
      }
 
      template<bool _MoveValue, typename _NodeGen>
        _Node_ptr
        _M_clone_node(_Node_ptr __x, _NodeGen& __node_gen)
        {
#if __cplusplus >= 201103L
          using _Vp = __conditional_t<_MoveValue,
                                      value_type&&,
                                      const value_type&>;
#endif
          _Node_ptr __tmp
            = __node_gen(_GLIBCXX_FORWARD(_Vp, *__x->_M_valptr()));
          __tmp->_M_color = __x->_M_color;
          __tmp->_M_left = __tmp->_M_right = _Base_ptr();
          return __tmp;
        }
 
    protected:
      typedef typename _Node_traits::_Header_t          _Header_t;
 
#if _GLIBCXX_INLINE_VERSION
      template<typename _Key_compare>
#else
      // Unused _Is_pod_comparator is kept as it is part of mangled name.
      template<typename _Key_compare,
               bool /* _Is_pod_comparator */ = __is_pod(_Key_compare)>
#endif
        struct _Rb_tree_impl
        : public _Node_allocator
        , public _Rb_tree_key_compare<_Key_compare>
        , public _Header_t
        {
          typedef _Rb_tree_key_compare<_Key_compare> _Base_key_compare;
 
          _Rb_tree_impl()
            _GLIBCXX_NOEXCEPT_IF(
                is_nothrow_default_constructible<_Node_allocator>::value
                && is_nothrow_default_constructible<_Base_key_compare>::value )
          : _Node_allocator()
          { }
 
          _Rb_tree_impl(const _Rb_tree_impl& __x)
          : _Node_allocator(_Node_alloc_traits::_S_select_on_copy(__x))
          , _Base_key_compare(__x._M_key_compare)
          , _Header_t()
          { }
 
#if __cplusplus < 201103L
          _Rb_tree_impl(const _Key_compare& __comp, const _Node_allocator& __a)
          : _Node_allocator(__a), _Base_key_compare(__comp)
          { }
#else
          _Rb_tree_impl(_Rb_tree_impl&&)
            noexcept( is_nothrow_move_constructible<_Base_key_compare>::value )
          = default;
 
          explicit
          _Rb_tree_impl(_Node_allocator&& __a)
          : _Node_allocator(std::move(__a))
          { }
 
          _Rb_tree_impl(_Rb_tree_impl&& __x, _Node_allocator&& __a)
          : _Node_allocator(std::move(__a)),
            _Base_key_compare(std::move(__x)),
            _Header_t(std::move(__x))
          { }
 
          _Rb_tree_impl(const _Key_compare& __comp, _Node_allocator&& __a)
          : _Node_allocator(std::move(__a)), _Base_key_compare(__comp)
          { }
#endif
        };
 
      _Rb_tree_impl<_Compare> _M_impl;
 
    protected:
      _Base_ptr&
      _M_root() _GLIBCXX_NOEXCEPT
      { return this->_M_impl._M_header._M_parent; }
 
      _Base_ptr
      _M_root() const _GLIBCXX_NOEXCEPT
      { return this->_M_impl._M_header._M_parent; }
 
      _Base_ptr&
      _M_leftmost() _GLIBCXX_NOEXCEPT
      { return this->_M_impl._M_header._M_left; }
 
      _Base_ptr
      _M_leftmost() const _GLIBCXX_NOEXCEPT
      { return this->_M_impl._M_header._M_left; }
 
      _Base_ptr&
      _M_rightmost() _GLIBCXX_NOEXCEPT
      { return this->_M_impl._M_header._M_right; }
 
      _Base_ptr
      _M_rightmost() const _GLIBCXX_NOEXCEPT
      { return this->_M_impl._M_header._M_right; }
 
      _Base_ptr
      _M_begin() const _GLIBCXX_NOEXCEPT
      { return this->_M_impl._M_header._M_parent; }
 
      _Node_ptr
      _M_begin_node() const _GLIBCXX_NOEXCEPT
      {
        _Base_ptr __begin = this->_M_impl._M_header._M_parent;
        return __begin
          ? static_cast<_Node&>(*__begin)._M_node_ptr()
          : _Node_ptr();
      }
 
      _Base_ptr
      _M_end() const _GLIBCXX_NOEXCEPT
      { return this->_M_impl._M_header._M_base_ptr(); }
 
      // _GLIBCXX_RESOLVE_LIB_DEFECTS
      // 2542. Missing const requirements for associative containers
      template<typename _Key1, typename _Key2>
        bool
        _M_key_compare(const _Key1& __k1, const _Key2& __k2) const
        {
#if __cplusplus >= 201103L
          // Enforce this here with a user-friendly message.
          static_assert(
            __is_invocable<const _Compare&, const _Key&, const _Key&>::value,
            "comparison object must be invocable with arguments of key_type"
          );
#endif
          return _M_impl._M_key_compare(__k1, __k2);
        }
 
      static const _Key&
      _S_key(const _Node& __node)
      { return _KeyOfValue()(*__node._M_valptr()); }
 
      static const _Key&
      _S_key(_Base_ptr __x)
      { return _S_key(static_cast<const _Node&>(*__x)); }
 
      static const _Key&
      _S_key(_Node_ptr __x)
      { return _S_key(*__x); }
 
      static _Base_ptr
      _S_left(_Base_ptr __x) _GLIBCXX_NOEXCEPT
      { return __x->_M_left; }
 
      static _Node_ptr
      _S_left(_Node_ptr __x)
      {
        return __x->_M_left
          ? static_cast<_Node&>(*__x->_M_left)._M_node_ptr()
          : _Node_ptr();
      }
 
      static _Base_ptr
      _S_right(_Base_ptr __x) _GLIBCXX_NOEXCEPT
      { return __x->_M_right; }
 
      static _Node_ptr
      _S_right(_Node_ptr __x) _GLIBCXX_NOEXCEPT
      {
        return __x->_M_right
          ? static_cast<_Node&>(*__x->_M_right)._M_node_ptr()
          : _Node_ptr();
      }
 
    public:
      typedef typename _Node_traits::_Iterator          iterator;
      typedef typename _Node_traits::_Const_iterator    const_iterator;
 
      typedef std::reverse_iterator<iterator>       reverse_iterator;
      typedef std::reverse_iterator<const_iterator> const_reverse_iterator;
 
#ifdef __glibcxx_node_extract // >= C++17
      using node_type = _Node_handle<_Key, _Val, _Node_allocator>;
      using insert_return_type = _Node_insert_return<
        __conditional_t<is_same_v<_Key, _Val>, const_iterator, iterator>,
        node_type>;
#endif
 
      pair<_Base_ptr, _Base_ptr>
      _M_get_insert_unique_pos(const key_type& __k);
 
      pair<_Base_ptr, _Base_ptr>
      _M_get_insert_equal_pos(const key_type& __k);
 
      pair<_Base_ptr, _Base_ptr>
      _M_get_insert_hint_unique_pos(const_iterator __pos,
                                    const key_type& __k);
 
      pair<_Base_ptr, _Base_ptr>
      _M_get_insert_hint_equal_pos(const_iterator __pos,
                                   const key_type& __k);
 
#ifdef __glibcxx_associative_heterogeneous_insertion  // C++26
      template <typename... _Args>
        iterator
        _M_emplace_here(bool __place_left, _Base_ptr __node, _Args&&... __args);
 
      template <typename _Kt>
        pair<_Base_ptr, _Base_ptr>
        _M_get_insert_unique_pos_tr(const _Kt& __k);
 
      template <typename _Kt>
        pair<_Base_ptr, _Base_ptr>
        _M_get_insert_hint_unique_pos_tr(const_iterator, const _Kt& __k);
#endif
 
    private:
#if __cplusplus >= 201103L
      template<typename _Arg, typename _NodeGen>
        iterator
        _M_insert_(_Base_ptr __x, _Base_ptr __y, _Arg&& __v, _NodeGen&);
 
      iterator
      _M_insert_node(_Base_ptr __x, _Base_ptr __y, _Node_ptr __z);
 
      template<typename _Arg>
        iterator
        _M_insert_lower(_Base_ptr __y, _Arg&& __v);
 
      template<typename _Arg>
        iterator
        _M_insert_equal_lower(_Arg&& __x);
 
      iterator
      _M_insert_lower_node(_Base_ptr __p, _Node_ptr __z);
 
      iterator
      _M_insert_equal_lower_node(_Node_ptr __z);
#else
      template<typename _NodeGen>
        iterator
        _M_insert_(_Base_ptr __x, _Base_ptr __y,
                   const value_type& __v, _NodeGen&);
 
      // _GLIBCXX_RESOLVE_LIB_DEFECTS
      // 233. Insertion hints in associative containers.
      iterator
      _M_insert_lower(_Base_ptr __y, const value_type& __v);
 
      iterator
      _M_insert_equal_lower(const value_type& __x);
#endif
 
      enum { __as_lvalue, __as_rvalue };
 
      template<bool _MoveValues, typename _NodeGen>
        _Base_ptr
        _M_copy(_Node_ptr, _Base_ptr, _NodeGen&);
 
      template<bool _MoveValues, typename _NodeGen>
        _Base_ptr
        _M_copy(const _Rb_tree& __x, _NodeGen& __gen)
        {
          _Base_ptr __root =
            _M_copy<_MoveValues>(__x._M_begin_node(), _M_end(), __gen);
          _M_leftmost() = _Node_base::_S_minimum(__root);
          _M_rightmost() = _Node_base::_S_maximum(__root);
          _M_impl._M_node_count = __x._M_impl._M_node_count;
          return __root;
        }
 
      _Base_ptr
      _M_copy(const _Rb_tree& __x)
      {
        _Alloc_node __an(*this);
        return _M_copy<__as_lvalue>(__x, __an);
      }
 
      void
      _M_erase(_Node_ptr __x);
 
      _Base_ptr
      _M_lower_bound(_Base_ptr __x, _Base_ptr __y,
                     const _Key& __k) const;
 
      template <typename _Kt>
        _Base_ptr
        _M_lower_bound_tr(_Base_ptr __x, _Base_ptr __y, const _Kt& __k) const;
 
      _Base_ptr
      _M_upper_bound(_Base_ptr __x, _Base_ptr __y,
                     const _Key& __k) const;
 
      template <typename _Kt>
        _Base_ptr
        _M_upper_bound_tr(_Base_ptr __x, _Base_ptr __y, const _Kt& __k) const;
 
    public:
      // allocation/deallocation
#if __cplusplus < 201103L
      _Rb_tree() { }
#else
      _Rb_tree() = default;
#endif
 
      _Rb_tree(const _Compare& __comp,
               const allocator_type& __a = allocator_type())
      : _M_impl(__comp, _Node_allocator(__a)) { }
 
      _Rb_tree(const _Rb_tree& __x)
      : _M_impl(__x._M_impl)
      {
        if (__x._M_root())
          _M_root() = _M_copy(__x);
      }
 
#if __cplusplus >= 201103L
      _Rb_tree(const allocator_type& __a)
      : _M_impl(_Node_allocator(__a))
      { }
 
      _Rb_tree(const _Rb_tree& __x, const allocator_type& __a)
      : _M_impl(__x._M_impl._M_key_compare, _Node_allocator(__a))
      {
        if (__x._M_root())
          _M_root() = _M_copy(__x);
      }
 
      _Rb_tree(_Rb_tree&&) = default;
 
      _Rb_tree(_Rb_tree&& __x, const allocator_type& __a)
      : _Rb_tree(std::move(__x), _Node_allocator(__a))
      { }
 
    private:
      _Rb_tree(_Rb_tree&& __x, _Node_allocator&& __a, true_type)
      noexcept(is_nothrow_default_constructible<_Compare>::value)
      : _M_impl(std::move(__x._M_impl), std::move(__a))
      { }
 
      _Rb_tree(_Rb_tree&& __x, _Node_allocator&& __a, false_type)
      : _M_impl(__x._M_impl._M_key_compare, std::move(__a))
      {
        if (__x._M_root())
          _M_move_data(__x, false_type{});
      }
 
    public:
      _Rb_tree(_Rb_tree&& __x, _Node_allocator&& __a)
      noexcept( noexcept(
        _Rb_tree(std::declval<_Rb_tree&&>(), std::declval<_Node_allocator&&>(),
                 std::declval<typename _Node_alloc_traits::is_always_equal>())) )
      : _Rb_tree(std::move(__x), std::move(__a),
                 typename _Node_alloc_traits::is_always_equal{})
      { }
#endif
 
      ~_Rb_tree() _GLIBCXX_NOEXCEPT
      { _M_erase(_M_begin_node()); }
 
      _Rb_tree&
      operator=(const _Rb_tree& __x);
 
      // Accessors.
      _Compare
      key_comp() const
      { return _M_impl._M_key_compare; }
 
      iterator
      begin() _GLIBCXX_NOEXCEPT
      { return iterator(this->_M_impl._M_header._M_left); }
 
      const_iterator
      begin() const _GLIBCXX_NOEXCEPT
      { return const_iterator(this->_M_impl._M_header._M_left); }
 
      iterator
      end() _GLIBCXX_NOEXCEPT
      { return iterator(_M_end()); }
 
      const_iterator
      end() const _GLIBCXX_NOEXCEPT
      { return const_iterator(_M_end()); }
 
      reverse_iterator
      rbegin() _GLIBCXX_NOEXCEPT
      { return reverse_iterator(end()); }
 
      const_reverse_iterator
      rbegin() const _GLIBCXX_NOEXCEPT
      { return const_reverse_iterator(end()); }
 
      reverse_iterator
      rend() _GLIBCXX_NOEXCEPT
      { return reverse_iterator(begin()); }
 
      const_reverse_iterator
      rend() const _GLIBCXX_NOEXCEPT
      { return const_reverse_iterator(begin()); }
 
      _GLIBCXX_NODISCARD bool
      empty() const _GLIBCXX_NOEXCEPT
      { return _M_impl._M_node_count == 0; }
 
      size_type
      size() const _GLIBCXX_NOEXCEPT
      { return _M_impl._M_node_count; }
 
      size_type
      max_size() const _GLIBCXX_NOEXCEPT
      { return _Node_alloc_traits::max_size(_M_get_Node_allocator()); }
 
      void
      swap(_Rb_tree& __t)
      _GLIBCXX_NOEXCEPT_IF(__is_nothrow_swappable<_Compare>::value);
 
      // Insert/erase.
#if __cplusplus >= 201103L
      template<typename _Arg>
        pair<iterator, bool>
        _M_insert_unique(_Arg&& __x);
 
      template<typename _Arg>
        iterator
        _M_insert_equal(_Arg&& __x);
 
      template<typename _Arg, typename _NodeGen>
        iterator
        _M_insert_unique_(const_iterator __pos, _Arg&& __x, _NodeGen&);
 
      template<typename _Arg>
        iterator
        _M_insert_unique_(const_iterator __pos, _Arg&& __x)
        {
          _Alloc_node __an(*this);
          return _M_insert_unique_(__pos, std::forward<_Arg>(__x), __an);
        }
 
      template<typename _Arg, typename _NodeGen>
        iterator
        _M_insert_equal_(const_iterator __pos, _Arg&& __x, _NodeGen&);
 
      template<typename _Arg>
        iterator
        _M_insert_equal_(const_iterator __pos, _Arg&& __x)
        {
          _Alloc_node __an(*this);
          return _M_insert_equal_(__pos, std::forward<_Arg>(__x), __an);
        }
 
      template<typename... _Args>
        pair<iterator, bool>
        _M_emplace_unique(_Args&&... __args);
 
      template<typename... _Args>
        iterator
        _M_emplace_equal(_Args&&... __args);
 
      template<typename... _Args>
        iterator
        _M_emplace_hint_unique(const_iterator __pos, _Args&&... __args);
 
      template<typename... _Args>
        iterator
        _M_emplace_hint_equal(const_iterator __pos, _Args&&... __args);
 
      template<typename _Iter>
        using __same_value_type
          = is_same<value_type, typename iterator_traits<_Iter>::value_type>;
 
      template<typename _InputIterator>
        __enable_if_t<__same_value_type<_InputIterator>::value>
        _M_insert_range_unique(_InputIterator __first, _InputIterator __last)
        {
          _Alloc_node __an(*this);
          for (; __first != __last; ++__first)
            _M_insert_unique_(end(), *__first, __an);
        }
 
      template<typename _InputIterator>
        __enable_if_t<!__same_value_type<_InputIterator>::value>
        _M_insert_range_unique(_InputIterator __first, _InputIterator __last)
        {
          for (; __first != __last; ++__first)
            _M_emplace_unique(*__first);
        }
 
      template<typename _InputIterator>
        __enable_if_t<__same_value_type<_InputIterator>::value>
        _M_insert_range_equal(_InputIterator __first, _InputIterator __last)
        {
          _Alloc_node __an(*this);
          for (; __first != __last; ++__first)
            _M_insert_equal_(end(), *__first, __an);
        }
 
      template<typename _InputIterator>
        __enable_if_t<!__same_value_type<_InputIterator>::value>
        _M_insert_range_equal(_InputIterator __first, _InputIterator __last)
        {
          for (; __first != __last; ++__first)
            _M_emplace_equal(*__first);
        }
#else
      pair<iterator, bool>
      _M_insert_unique(const value_type& __x);
 
      iterator
      _M_insert_equal(const value_type& __x);
 
      template<typename _NodeGen>
        iterator
        _M_insert_unique_(const_iterator __pos, const value_type& __x,
                          _NodeGen&);
 
      iterator
      _M_insert_unique_(const_iterator __pos, const value_type& __x)
      {
        _Alloc_node __an(*this);
        return _M_insert_unique_(__pos, __x, __an);
      }
 
      template<typename _NodeGen>
        iterator
        _M_insert_equal_(const_iterator __pos, const value_type& __x,
                         _NodeGen&);
      iterator
      _M_insert_equal_(const_iterator __pos, const value_type& __x)
      {
        _Alloc_node __an(*this);
        return _M_insert_equal_(__pos, __x, __an);
      }
 
      template<typename _InputIterator>
        void
        _M_insert_range_unique(_InputIterator __first, _InputIterator __last)
        {
          _Alloc_node __an(*this);
          for (; __first != __last; ++__first)
            _M_insert_unique_(end(), *__first, __an);
        }
 
      template<typename _InputIterator>
        void
        _M_insert_range_equal(_InputIterator __first, _InputIterator __last)
        {
          _Alloc_node __an(*this);
          for (; __first != __last; ++__first)
            _M_insert_equal_(end(), *__first, __an);
        }
#endif
 
    private:
      void
      _M_erase_aux(const_iterator __position);
 
      void
      _M_erase_aux(const_iterator __first, const_iterator __last);
 
    public:
#if __cplusplus >= 201103L
      // _GLIBCXX_RESOLVE_LIB_DEFECTS
      // DR 130. Associative erase should return an iterator.
      _GLIBCXX_ABI_TAG_CXX11
      iterator
      erase(const_iterator __position)
      {
        __glibcxx_assert(__position != end());
        const_iterator __result = __position;
        ++__result;
        _M_erase_aux(__position);
        return iterator(__result._M_node);
      }
 
      // LWG 2059.
      _GLIBCXX_ABI_TAG_CXX11
      iterator
      erase(iterator __position)
      {
        __glibcxx_assert(__position != end());
        iterator __result = __position;
        ++__result;
        _M_erase_aux(__position);
        return __result;
      }
#else
      void
      erase(iterator __position)
      {
        __glibcxx_assert(__position != end());
        _M_erase_aux(__position);
      }
 
      void
      erase(const_iterator __position)
      {
        __glibcxx_assert(__position != end());
        _M_erase_aux(__position);
      }
#endif
 
      size_type
      erase(const key_type& __x);
 
      template <typename _Kt>
        size_type
        _M_erase_tr(const _Kt& __x);
 
      size_type
      _M_erase_unique(const key_type& __x);
 
#if __cplusplus >= 201103L
      // _GLIBCXX_RESOLVE_LIB_DEFECTS
      // DR 130. Associative erase should return an iterator.
      _GLIBCXX_ABI_TAG_CXX11
      iterator
      erase(const_iterator __first, const_iterator __last)
      {
        _M_erase_aux(__first, __last);
        return iterator(__last._M_node);
      }
#else
      void
      erase(iterator __first, iterator __last)
      { _M_erase_aux(__first, __last); }
 
      void
      erase(const_iterator __first, const_iterator __last)
      { _M_erase_aux(__first, __last); }
#endif
 
      void
      clear() _GLIBCXX_NOEXCEPT
      {
        _M_erase(_M_begin_node());
        _M_impl._M_reset();
      }
 
      // Set operations.
      iterator
      find(const key_type& __k);
 
      const_iterator
      find(const key_type& __k) const;
 
      size_type
      count(const key_type& __k) const;
 
      iterator
      lower_bound(const key_type& __k)
      { return iterator(_M_lower_bound(_M_begin(), _M_end(), __k)); }
 
      const_iterator
      lower_bound(const key_type& __k) const
      {
        return const_iterator
          (_M_lower_bound(_M_begin(), _M_end(), __k));
      }
 
      iterator
      upper_bound(const key_type& __k)
      { return iterator(_M_upper_bound(_M_begin(), _M_end(), __k)); }
 
      const_iterator
      upper_bound(const key_type& __k) const
      {
        return const_iterator
          (_M_upper_bound(_M_begin(), _M_end(), __k));
      }
 
      pair<iterator, iterator>
      equal_range(const key_type& __k);
 
      pair<const_iterator, const_iterator>
      equal_range(const key_type& __k) const;
 
#ifdef __glibcxx_generic_associative_lookup // C++ >= 14
      template<typename _Kt,
               typename _Req = __has_is_transparent_t<_Compare, _Kt>>
        iterator
        _M_find_tr(const _Kt& __k)
        {
          const _Rb_tree* __const_this = this;
          return iterator(__const_this->_M_find_tr(__k)._M_node);
        }
 
      template<typename _Kt,
               typename _Req = __has_is_transparent_t<_Compare, _Kt>>
        const_iterator
        _M_find_tr(const _Kt& __k) const
        {
          const_iterator __j(_M_lower_bound_tr(__k));
          if (__j != end() && _M_key_compare(__k, _S_key(__j._M_node)))
            __j = end();
          return __j;
        }
 
      template<typename _Kt,
               typename _Req = __has_is_transparent_t<_Compare, _Kt>>
        size_type
        _M_count_tr(const _Kt& __k) const
        {
          auto __p = _M_equal_range_tr(__k);
          return std::distance(__p.first, __p.second);
        }
 
      template<typename _Kt,
               typename _Req = __has_is_transparent_t<_Compare, _Kt>>
        _Base_ptr
        _M_lower_bound_tr(const _Kt& __k) const
        {
          auto __x = _M_begin();
          auto __y = _M_end();
          while (__x)
            if (!_M_key_compare(_S_key(__x), __k))
              {
                __y = __x;
                __x = _S_left(__x);
              }
            else
              __x = _S_right(__x);
          return __y;
        }
 
      template<typename _Kt,
               typename _Req = __has_is_transparent_t<_Compare, _Kt>>
        _Base_ptr
        _M_upper_bound_tr(const _Kt& __k) const
        {
          auto __x = _M_begin();
          auto __y = _M_end();
          while (__x)
            if (_M_key_compare(__k, _S_key(__x)))
              {
                __y = __x;
                __x = _S_left(__x);
              }
            else
              __x = _S_right(__x);
          return __y;
        }
 
      template<typename _Kt,
               typename _Req = __has_is_transparent_t<_Compare, _Kt>>
        pair<iterator, iterator>
        _M_equal_range_tr(const _Kt& __k)
        {
          const _Rb_tree* __const_this = this;
          auto __ret = __const_this->_M_equal_range_tr(__k);
          return
            { iterator(__ret.first._M_node), iterator(__ret.second._M_node) };
        }
 
      template<typename _Kt,
               typename _Req = __has_is_transparent_t<_Compare, _Kt>>
        pair<const_iterator, const_iterator>
        _M_equal_range_tr(const _Kt& __k) const
        {
          auto __x = _M_begin();
          auto __y = _M_end();
          while (__x)
            {
              if (_M_key_compare(_S_key(__x), __k))
                __x = _S_right(__x);
              else if (_M_key_compare(__k, _S_key(__x)))
                {
                  __y = __x;
                  __x = _S_left(__x);
                }
              else
                {
                  auto __xu(__x);
                  auto __yu(__y);
                  __y = __x;
                  __x = _S_left(__x);
                  __xu = _S_right(__xu);
                  return { const_iterator(_M_lower_bound_tr(__x, __y, __k)),
                         const_iterator(_M_upper_bound_tr(__xu, __yu, __k)) };
                }
            }
          return { const_iterator(__y), const_iterator(__y) };
        }
#endif // __glibcxx_generic_associative_lookup
 
      // Debugging.
      bool
      __rb_verify() const;
 
#if __cplusplus >= 201103L
      _Rb_tree&
      operator=(_Rb_tree&&)
      noexcept(_Node_alloc_traits::_S_nothrow_move()
               && is_nothrow_move_assignable<_Compare>::value);
 
      template<typename _Iterator>
        void
        _M_assign_unique(_Iterator, _Iterator);
 
      template<typename _Iterator>
        void
        _M_assign_equal(_Iterator, _Iterator);
 
    private:
      // Move elements from container with equal allocator.
      void
      _M_move_data(_Rb_tree& __x, true_type)
      { _M_impl._M_move_data(__x._M_impl); }
 
      // Move elements from container with possibly non-equal allocator,
      // which might result in a copy not a move.
      void
      _M_move_data(_Rb_tree&, false_type);
 
      // Move assignment from container with equal allocator.
      void
      _M_move_assign(_Rb_tree&, true_type);
 
      // Move assignment from container with possibly non-equal allocator,
      // which might result in a copy not a move.
      void
      _M_move_assign(_Rb_tree&, false_type);
#endif
 
#ifdef __glibcxx_node_extract // >= C++17
      static _Node_ptr
      _S_adapt(typename _Node_alloc_traits::pointer __ptr)
      {
#if _GLIBCXX_USE_ALLOC_PTR_FOR_RB_TREE
        return __ptr;
#else
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wc++17-extensions" // if constexpr
        using __alloc_ptr = typename _Node_alloc_traits::pointer;
        if constexpr (is_same<_Node_ptr, __alloc_ptr>::value)
          return __ptr;
        else
          return std::__to_address(__ptr);
#pragma GCC diagnostic pop
#endif
      }
 
    public:
      /// Re-insert an extracted node.
      insert_return_type
      _M_reinsert_node_unique(node_type&& __nh)
      {
        insert_return_type __ret;
        if (__nh.empty())
          __ret.position = end();
        else
          {
            __glibcxx_assert(_M_get_Node_allocator() == *__nh._M_alloc);
 
            auto __res = _M_get_insert_unique_pos(__nh._M_key());
            if (__res.second)
              {
                __ret.position
                  = _M_insert_node(__res.first, __res.second,
                                   _S_adapt(__nh._M_ptr));
                __nh.release();
                __ret.inserted = true;
              }
            else
              {
                __ret.node = std::move(__nh);
                __ret.position = iterator(__res.first);
                __ret.inserted = false;
              }
          }
        return __ret;
      }
 
      /// Re-insert an extracted node.
      iterator
      _M_reinsert_node_equal(node_type&& __nh)
      {
        iterator __ret;
        if (__nh.empty())
          __ret = end();
        else
          {
            __glibcxx_assert(_M_get_Node_allocator() == *__nh._M_alloc);
            auto __res = _M_get_insert_equal_pos(__nh._M_key());
            if (__res.second)
              __ret = _M_insert_node(__res.first, __res.second,
                                     _S_adapt(__nh._M_ptr));
            else
              __ret = _M_insert_equal_lower_node(_S_adapt(__nh._M_ptr));
            __nh.release();
          }
        return __ret;
      }
 
      /// Re-insert an extracted node.
      iterator
      _M_reinsert_node_hint_unique(const_iterator __hint, node_type&& __nh)
      {
        iterator __ret;
        if (__nh.empty())
          __ret = end();
        else
          {
            __glibcxx_assert(_M_get_Node_allocator() == *__nh._M_alloc);
            auto __res = _M_get_insert_hint_unique_pos(__hint, __nh._M_key());
            if (__res.second)
              {
                __ret = _M_insert_node(__res.first, __res.second,
                                       _S_adapt(__nh._M_ptr));
                __nh.release();
              }
            else
              __ret = iterator(__res.first);
          }
        return __ret;
      }
 
      /// Re-insert an extracted node.
      iterator
      _M_reinsert_node_hint_equal(const_iterator __hint, node_type&& __nh)
      {
        iterator __ret;
        if (__nh.empty())
          __ret = end();
        else
          {
            __glibcxx_assert(_M_get_Node_allocator() == *__nh._M_alloc);
            auto __res = _M_get_insert_hint_equal_pos(__hint, __nh._M_key());
            if (__res.second)
              __ret = _M_insert_node(__res.first, __res.second,
                                     _S_adapt(__nh._M_ptr));
            else
              __ret = _M_insert_equal_lower_node(_S_adapt(__nh._M_ptr));
            __nh.release();
          }
        return __ret;
      }
 
      /// Extract a node.
      node_type
      extract(const_iterator __pos)
      {
        auto __ptr = _Node_traits::_S_rebalance_for_erase
          (__pos._M_node, _M_impl._M_header);
        --_M_impl._M_node_count;
        auto __node_ptr = static_cast<_Node&>(*__ptr)._M_node_ptr();
#if _GLIBCXX_USE_ALLOC_PTR_FOR_RB_TREE
        return { __node_ptr, _M_get_Node_allocator() };
#else
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wc++17-extensions" // if constexpr
        using __alloc_ptr = typename _Node_alloc_traits::pointer;
        if constexpr (is_same<_Node_ptr, __alloc_ptr>::value)
          return { __node_ptr, _M_get_Node_allocator() };
        else
          {
            auto __ap = pointer_traits<__alloc_ptr>::pointer_to(*__node_ptr);
            return { __ap, _M_get_Node_allocator() };
          }
#pragma GCC diagnostic pop
#endif
      }
 
      /// Extract a node.
      node_type
      extract(const key_type& __k)
      {
        node_type __nh;
        auto __pos = find(__k);
        if (__pos != end())
          __nh = extract(const_iterator(__pos));
        return __nh;
      }
 
      template <typename _Kt>
        node_type
        _M_extract_tr(const _Kt& __k)
        {
          node_type __nh;
          auto __pos = _M_find_tr(__k);
          if (__pos != end())
            __nh = extract(const_iterator(__pos));
          return __nh;
        }
 
      template<typename _Compare2>
        using _Compatible_tree
          = _Rb_tree<_Key, _Val, _KeyOfValue, _Compare2, _Alloc>;
 
      template<typename, typename>
        friend struct _Rb_tree_merge_helper;
 
      /// Merge from a compatible container into one with unique keys.
      template<typename _Compare2>
        void
        _M_merge_unique(_Compatible_tree<_Compare2>& __src) noexcept
        {
          using _Merge_helper = _Rb_tree_merge_helper<_Rb_tree, _Compare2>;
          for (auto __i = __src.begin(), __end = __src.end(); __i != __end;)
            {
              auto __pos = __i++;
              auto __res = _M_get_insert_unique_pos(_KeyOfValue()(*__pos));
              if (__res.second)
                {
                  auto& __src_impl = _Merge_helper::_S_get_impl(__src);
                  auto __ptr = _Node_traits::_S_rebalance_for_erase
                    (__pos._M_node, __src_impl._M_header);
                  --__src_impl._M_node_count;
                  auto __node_ptr = static_cast<_Node&>(*__ptr)._M_node_ptr();
                  _M_insert_node(__res.first, __res.second, __node_ptr);
                }
            }
        }
 
      /// Merge from a compatible container into one with equivalent keys.
      template<typename _Compare2>
        void
        _M_merge_equal(_Compatible_tree<_Compare2>& __src) noexcept
        {
          using _Merge_helper = _Rb_tree_merge_helper<_Rb_tree, _Compare2>;
          for (auto __i = __src.begin(), __end = __src.end(); __i != __end;)
            {
              auto __pos = __i++;
              auto __res = _M_get_insert_equal_pos(_KeyOfValue()(*__pos));
              if (__res.second)
                {
                  auto& __src_impl = _Merge_helper::_S_get_impl(__src);
                  auto __ptr = _Node_traits::_S_rebalance_for_erase
                    (__pos._M_node, __src_impl._M_header);
                  --__src_impl._M_node_count;
                  auto __node_ptr = static_cast<_Node&>(*__ptr)._M_node_ptr();
                  _M_insert_node(__res.first, __res.second, __node_ptr);
                }
            }
        }
#endif // C++17 node_extract
 
      friend bool
      operator==(const _Rb_tree& __x, const _Rb_tree& __y)
      {
        return __x.size() == __y.size()
          && std::equal(__x.begin(), __x.end(), __y.begin());
      }
 
#if __cpp_lib_three_way_comparison
      friend auto
      operator<=>(const _Rb_tree& __x, const _Rb_tree& __y)
      {
        if constexpr (requires { typename __detail::__synth3way_t<_Val>; })
          return std::lexicographical_compare_three_way(__x.begin(), __x.end(),
                                                        __y.begin(), __y.end(),
                                                        __detail::__synth3way);
      }
#else
      friend bool
      operator<(const _Rb_tree& __x, const _Rb_tree& __y)
      {
        return std::lexicographical_compare(__x.begin(), __x.end(),
                                            __y.begin(), __y.end());
      }
#endif
 
    private:
#if __cplusplus >= 201103L
      // An RAII _Node handle
      struct _Auto_node
      {
        template<typename... _Args>
          _Auto_node(_Rb_tree& __t, _Args&&... __args)
          : _M_t(__t),
            _M_node(__t._M_create_node(std::forward<_Args>(__args)...))
          { }
 
        ~_Auto_node()
        {
          if (_M_node)
            _M_t._M_drop_node(_M_node);
        }
 
        _Auto_node(_Auto_node&& __n)
        : _M_t(__n._M_t), _M_node(__n._M_node)
        { __n._M_node = nullptr; }
 
        const _Key&
        _M_key() const
        { return _S_key(_M_node); }
 
        iterator
        _M_insert(pair<_Base_ptr, _Base_ptr> __p)
        {
          auto __it = _M_t._M_insert_node(__p.first, __p.second, _M_node);
          _M_node = nullptr;
          return __it;
        }
 
        iterator
        _M_insert_equal_lower()
        {
          auto __it = _M_t._M_insert_equal_lower_node(_M_node);
          _M_node = nullptr;
          return __it;
        }
 
        _Rb_tree& _M_t;
        _Node_ptr _M_node;
      };
#endif // C++11
    };
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
    inline void
    swap(_Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>& __x,
         _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>& __y)
    { __x.swap(__y); }
 
#if __cplusplus >= 201103L
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
    void
    _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
    _M_move_data(_Rb_tree& __x, false_type)
    {
      if (_M_get_Node_allocator() == __x._M_get_Node_allocator())
        _M_move_data(__x, true_type());
      else
        {
          constexpr bool __move = !__move_if_noexcept_cond<value_type>::value;
          _Alloc_node __an(*this);
          _M_root() = _M_copy<__move>(__x, __an);
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wc++17-extensions" // if constexpr
          if constexpr (__move)
            __x.clear();
#pragma GCC diagnostic pop
        }
    }
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
    inline void
    _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
    _M_move_assign(_Rb_tree& __x, true_type)
    {
      clear();
      if (__x._M_root())
        _M_move_data(__x, true_type());
      std::__alloc_on_move(_M_get_Node_allocator(),
                           __x._M_get_Node_allocator());
    }
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
    void
    _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
    _M_move_assign(_Rb_tree& __x, false_type)
    {
      if (_M_get_Node_allocator() == __x._M_get_Node_allocator())
        return _M_move_assign(__x, true_type{});
 
      // Try to move each node reusing existing nodes and copying __x nodes
      // structure.
      _Reuse_or_alloc_node __roan(*this);
      _M_impl._M_reset();
      if (__x._M_root())
        {
          _M_root() = _M_copy<__as_rvalue>(__x, __roan);
          __x.clear();
        }
    }
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
    inline _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>&
    _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
    operator=(_Rb_tree&& __x)
    noexcept(_Node_alloc_traits::_S_nothrow_move()
             && is_nothrow_move_assignable<_Compare>::value)
    {
      _M_impl._M_key_compare = std::move(__x._M_impl._M_key_compare);
      _M_move_assign(__x,
                     __bool_constant<_Node_alloc_traits::_S_nothrow_move()>());
      return *this;
    }
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
    template<typename _Iterator>
      void
      _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
      _M_assign_unique(_Iterator __first, _Iterator __last)
      {
        _Reuse_or_alloc_node __roan(*this);
        _M_impl._M_reset();
        for (; __first != __last; ++__first)
          _M_insert_unique_(end(), *__first, __roan);
      }
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
    template<typename _Iterator>
      void
      _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
      _M_assign_equal(_Iterator __first, _Iterator __last)
      {
        _Reuse_or_alloc_node __roan(*this);
        _M_impl._M_reset();
        for (; __first != __last; ++__first)
          _M_insert_equal_(end(), *__first, __roan);
      }
#endif // C++11
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
    _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>&
    _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
    operator=(const _Rb_tree& __x)
    {
      if (this != std::__addressof(__x))
        {
          // Note that _Key may be a constant type.
#if __cplusplus >= 201103L
          if (_Node_alloc_traits::_S_propagate_on_copy_assign())
            {
              auto& __this_alloc = this->_M_get_Node_allocator();
              auto& __that_alloc = __x._M_get_Node_allocator();
              if (!_Node_alloc_traits::_S_always_equal()
                  && __this_alloc != __that_alloc)
                {
                  // Replacement allocator cannot free existing storage, we need
                  // to erase nodes first.
                  clear();
                  std::__alloc_on_copy(__this_alloc, __that_alloc);
                }
            }
#endif
 
          _Reuse_or_alloc_node __roan(*this);
          _M_impl._M_reset();
          _M_impl._M_key_compare = __x._M_impl._M_key_compare;
          if (__x._M_root())
            _M_root() = _M_copy<__as_lvalue>(__x, __roan);
        }
 
      return *this;
    }
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
#if __cplusplus >= 201103L
    template<typename _Arg, typename _NodeGen>
#else
    template<typename _NodeGen>
#endif
      typename _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::iterator
      _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
      _M_insert_(_Base_ptr __x, _Base_ptr __p,
#if __cplusplus >= 201103L
                 _Arg&& __v,
#else
                 const _Val& __v,
#endif
                 _NodeGen& __node_gen)
      {
        bool __insert_left = (__x || __p == _M_end()
                              || _M_key_compare(_KeyOfValue()(__v),
                                                _S_key(__p)));
 
        _Base_ptr __z =
          __node_gen(_GLIBCXX_FORWARD(_Arg, __v))->_M_base_ptr();
 
        _Node_traits::_S_insert_and_rebalance
          (__insert_left, __z, __p, this->_M_impl._M_header);
        ++_M_impl._M_node_count;
        return iterator(__z);
      }
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
#if __cplusplus >= 201103L
    template<typename _Arg>
#endif
    typename _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::iterator
    _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
#if __cplusplus >= 201103L
    _M_insert_lower(_Base_ptr __p, _Arg&& __v)
#else
    _M_insert_lower(_Base_ptr __p, const _Val& __v)
#endif
    {
      bool __insert_left = (__p == _M_end()
                            || !_M_key_compare(_S_key(__p),
                                               _KeyOfValue()(__v)));
 
      _Base_ptr __z =
        _M_create_node(_GLIBCXX_FORWARD(_Arg, __v))->_M_base_ptr();
      _Node_traits::_S_insert_and_rebalance
        (__insert_left, __z, __p, this->_M_impl._M_header);
      ++_M_impl._M_node_count;
      return iterator(__z);
    }
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
#if __cplusplus >= 201103L
    template<typename _Arg>
#endif
    typename _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::iterator
    _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
#if __cplusplus >= 201103L
    _M_insert_equal_lower(_Arg&& __v)
#else
    _M_insert_equal_lower(const _Val& __v)
#endif
    {
      _Base_ptr __x = _M_begin();
      _Base_ptr __y = _M_end();
      while (__x)
        {
          __y = __x;
          __x = !_M_key_compare(_S_key(__x), _KeyOfValue()(__v)) ?
                _S_left(__x) : _S_right(__x);
        }
      return _M_insert_lower(__y, _GLIBCXX_FORWARD(_Arg, __v));
    }
 
  template<typename _Key, typename _Val, typename _KoV,
           typename _Compare, typename _Alloc>
    template<bool _MoveValues, typename _NodeGen>
      typename _Rb_tree<_Key, _Val, _KoV, _Compare, _Alloc>::_Base_ptr
      _Rb_tree<_Key, _Val, _KoV, _Compare, _Alloc>::
      _M_copy(_Node_ptr __x, _Base_ptr __p, _NodeGen& __node_gen)
      {
        // Structural copy. __x and __p must be non-null.
        _Node_ptr __top = _M_clone_node<_MoveValues>(__x, __node_gen);
        _Base_ptr __top_base = __top->_M_base_ptr();
        __top->_M_parent = __p;
 
        __try
          {
            if (__x->_M_right)
              __top->_M_right =
                _M_copy<_MoveValues>(_S_right(__x), __top_base, __node_gen);
            __p = __top_base;
            __x = _S_left(__x);
 
            while (__x)
              {
                _Base_ptr __y =
                  _M_clone_node<_MoveValues>(__x, __node_gen)->_M_base_ptr();
                __p->_M_left = __y;
                __y->_M_parent = __p;
                if (__x->_M_right)
                  __y->_M_right = _M_copy<_MoveValues>(_S_right(__x),
                                                       __y, __node_gen);
                __p = __y;
                __x = _S_left(__x);
              }
          }
        __catch(...)
          {
            _M_erase(__top);
            __throw_exception_again;
          }
        return __top_base;
      }
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
    void
    _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
    _M_erase(_Node_ptr __x)
    {
      // Erase without rebalancing.
      while (__x)
        {
          _M_erase(_S_right(__x));
          _Node_ptr __y = _S_left(__x);
          _M_drop_node(__x);
          __x = __y;
        }
    }
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
    typename _Rb_tree<_Key, _Val, _KeyOfValue,
                      _Compare, _Alloc>::_Base_ptr
    _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
    _M_lower_bound(_Base_ptr __x, _Base_ptr __y,
                   const _Key& __k) const
    {
      while (__x)
        if (!_M_key_compare(_S_key(__x), __k))
          __y = __x, __x = _S_left(__x);
        else
          __x = _S_right(__x);
      return __y;
    }
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
        typename _Compare, typename _Alloc>
    template <typename _Kt>
      typename _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::_Base_ptr
      _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
      _M_lower_bound_tr(_Base_ptr __x, _Base_ptr __y, const _Kt& __k) const
      {
        while (__x)
          if (!_M_key_compare(_S_key(__x), __k))
            __y = __x, __x = _S_left(__x);
          else
            __x = _S_right(__x);
        return __y;
      }
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
    typename _Rb_tree<_Key, _Val, _KeyOfValue,
                      _Compare, _Alloc>::_Base_ptr
    _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
    _M_upper_bound(_Base_ptr __x, _Base_ptr __y,
                   const _Key& __k) const
    {
      while (__x)
        if (_M_key_compare(__k, _S_key(__x)))
          __y = __x, __x = _S_left(__x);
        else
          __x = _S_right(__x);
      return __y;
    }
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
    template <typename _Kt>
      typename _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::_Base_ptr
      _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
      _M_upper_bound_tr(_Base_ptr __x, _Base_ptr __y, const _Kt& __k) const
      {
        while (__x)
          if (_M_key_compare(__k, _S_key(__x)))
            __y = __x, __x = _S_left(__x);
          else
            __x = _S_right(__x);
        return __y;
      }
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
    pair<typename _Rb_tree<_Key, _Val, _KeyOfValue,
                           _Compare, _Alloc>::iterator,
         typename _Rb_tree<_Key, _Val, _KeyOfValue,
                           _Compare, _Alloc>::iterator>
    _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
    equal_range(const _Key& __k)
    {
      typedef pair<iterator, iterator> _Ret;
 
      _Base_ptr __x = _M_begin();
      _Base_ptr __y = _M_end();
      while (__x)
        {
          if (_M_key_compare(_S_key(__x), __k))
            __x = _S_right(__x);
          else if (_M_key_compare(__k, _S_key(__x)))
            __y = __x, __x = _S_left(__x);
          else
            {
              _Base_ptr __xu(__x);
              _Base_ptr __yu(__y);
              __y = __x, __x = _S_left(__x);
              __xu = _S_right(__xu);
              return _Ret(iterator(_M_lower_bound(__x, __y, __k)),
                          iterator(_M_upper_bound(__xu, __yu, __k)));
            }
        }
      return _Ret(iterator(__y), iterator(__y));
    }
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
    pair<typename _Rb_tree<_Key, _Val, _KeyOfValue,
                           _Compare, _Alloc>::const_iterator,
         typename _Rb_tree<_Key, _Val, _KeyOfValue,
                           _Compare, _Alloc>::const_iterator>
    _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
    equal_range(const _Key& __k) const
    {
      typedef pair<const_iterator, const_iterator> _Ret;
 
      _Base_ptr __x = _M_begin();
      _Base_ptr __y = _M_end();
      while (__x)
        {
          if (_M_key_compare(_S_key(__x), __k))
            __x = _S_right(__x);
          else if (_M_key_compare(__k, _S_key(__x)))
            __y = __x, __x = _S_left(__x);
          else
            {
              _Base_ptr __xu(__x);
              _Base_ptr __yu(__y);
              __y = __x, __x = _S_left(__x);
              __xu = _S_right(__xu);
              return _Ret(const_iterator(_M_lower_bound(__x, __y, __k)),
                          const_iterator(_M_upper_bound(__xu, __yu, __k)));
            }
        }
      return _Ret(const_iterator(__y), const_iterator(__y));
    }
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
    void
    _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
    swap(_Rb_tree& __t)
    _GLIBCXX_NOEXCEPT_IF(__is_nothrow_swappable<_Compare>::value)
    {
      if (!_M_root())
        {
          if (__t._M_root())
            _M_impl._M_move_data(__t._M_impl);
        }
      else if (!__t._M_root())
        __t._M_impl._M_move_data(_M_impl);
      else
        {
          std::swap(_M_root(),__t._M_root());
          std::swap(_M_leftmost(),__t._M_leftmost());
          std::swap(_M_rightmost(),__t._M_rightmost());
 
          _M_root()->_M_parent = _M_end();
          __t._M_root()->_M_parent = __t._M_end();
          std::swap(this->_M_impl._M_node_count, __t._M_impl._M_node_count);
        }
      // No need to swap header's color as it does not change.
 
      using std::swap;
      swap(this->_M_impl._M_key_compare, __t._M_impl._M_key_compare);
 
      _Node_alloc_traits::_S_on_swap(_M_get_Node_allocator(),
                                     __t._M_get_Node_allocator());
    }
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
    pair<typename _Rb_tree<_Key, _Val, _KeyOfValue,
                           _Compare, _Alloc>::_Base_ptr,
         typename _Rb_tree<_Key, _Val, _KeyOfValue,
                           _Compare, _Alloc>::_Base_ptr>
    _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
    _M_get_insert_unique_pos(const key_type& __k)
    {
      typedef pair<_Base_ptr, _Base_ptr> _Res;
      _Base_ptr __x = _M_begin();
      _Base_ptr __y = _M_end();
      bool __comp = true;
      while (__x)
        {
          __y = __x;
          __comp = _M_key_compare(__k, _S_key(__x));
          __x = __comp ? _S_left(__x) : _S_right(__x);
        }
      iterator __j = iterator(__y);
      if (__comp)
        {
          if (__j == begin())
            return _Res(__x, __y);
          else
            --__j;
        }
      if (_M_key_compare(_S_key(__j._M_node), __k))
        return _Res(__x, __y);
      return _Res(__j._M_node, _Base_ptr());
    }
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
    pair<typename _Rb_tree<_Key, _Val, _KeyOfValue,
                           _Compare, _Alloc>::_Base_ptr,
         typename _Rb_tree<_Key, _Val, _KeyOfValue,
                           _Compare, _Alloc>::_Base_ptr>
    _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
    _M_get_insert_equal_pos(const key_type& __k)
    {
      typedef pair<_Base_ptr, _Base_ptr> _Res;
      _Base_ptr __x = _M_begin();
      _Base_ptr __y = _M_end();
      while (__x)
        {
          __y = __x;
          __x = _M_key_compare(__k, _S_key(__x)) ? _S_left(__x) : _S_right(__x);
        }
      return _Res(__x, __y);
    }
 
#ifdef __glibcxx_associative_heterogeneous_insertion  // C++26
 
  // Multiple elements may compare equal to __k. Identify the first
  // of any such elements, or insert normally.
 
  template <typename _Key, typename _Val, typename _KeyOfValue,
            typename _Compare, typename _Alloc>
    template <typename _Kt>
      auto
      _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
      _M_get_insert_unique_pos_tr(const _Kt& __k)
      -> pair<_Base_ptr, _Base_ptr>
      {
        if (size() == 0)
          return { _M_end(), _M_end() }; // Insert as root.
 
        _Base_ptr __x = _M_begin(), __y = __x;
        bool __k_le_y = false;
        do
          {
            __y = __x;
            __k_le_y = ! _M_key_compare(_S_key(__x), __k);
            __x = __k_le_y ? _S_left(__x) : _S_right(__x);
          }
        while (__x);
        // If !__k_le_y, __k > *__y;
        //   If __y is rightmost, put at _M_right under *__y.
        //   else if __k < *(__y+1), put at _M_right under *__y.
        //   else __k == *(__y+1), do not insert, report (__y+1).
        // else, __k_le_y, __k <= *__y;
        //   If __k < *__Y, put at _M_left under *__y.
        //   else __k == *__y, do not insert, report __y.
        auto __j = iterator(__y);
        if (! __k_le_y)  // k > *__y
          {
            if (__y == _M_rightmost())
              return { {}, __y };   // Place to right under __y.
            ++__j;
          }
        if (_M_key_compare(__k, _S_key(__j._M_node)))
          {
            if (__k_le_y)
              return { __y, __y };  // Place to left under __y.
            else
              return { {}, __y };   // Place to right under __y.
          }
        return { __j._M_node, {} };    // No insert.
      }
#endif
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
#if __cplusplus >= 201103L
    template<typename _Arg>
#endif
    pair<typename _Rb_tree<_Key, _Val, _KeyOfValue,
                           _Compare, _Alloc>::iterator, bool>
    _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
#if __cplusplus >= 201103L
    _M_insert_unique(_Arg&& __v)
#else
    _M_insert_unique(const _Val& __v)
#endif
    {
      typedef pair<iterator, bool> _Res;
      pair<_Base_ptr, _Base_ptr> __res
        = _M_get_insert_unique_pos(_KeyOfValue()(__v));
 
      if (__res.second)
        {
          _Alloc_node __an(*this);
          return _Res(_M_insert_(__res.first, __res.second,
                                 _GLIBCXX_FORWARD(_Arg, __v), __an),
                      true);
        }
 
      return _Res(iterator(__res.first), false);
    }
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
#if __cplusplus >= 201103L
    template<typename _Arg>
#endif
    typename _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::iterator
    _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
#if __cplusplus >= 201103L
    _M_insert_equal(_Arg&& __v)
#else
    _M_insert_equal(const _Val& __v)
#endif
    {
      pair<_Base_ptr, _Base_ptr> __res
        = _M_get_insert_equal_pos(_KeyOfValue()(__v));
      _Alloc_node __an(*this);
      return _M_insert_(__res.first, __res.second,
                        _GLIBCXX_FORWARD(_Arg, __v), __an);
    }
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
    pair<typename _Rb_tree<_Key, _Val, _KeyOfValue,
                           _Compare, _Alloc>::_Base_ptr,
         typename _Rb_tree<_Key, _Val, _KeyOfValue,
                           _Compare, _Alloc>::_Base_ptr>
    _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
    _M_get_insert_hint_unique_pos(const_iterator __position,
                                  const key_type& __k)
    {
      typedef pair<_Base_ptr, _Base_ptr> _Res;
 
      // end()
      if (__position._M_node == _M_end())
        {
          if (size() > 0 && _M_key_compare(_S_key(_M_rightmost()), __k))
            return _Res(_Base_ptr(), _M_rightmost());
          else
            return _M_get_insert_unique_pos(__k);
        }
      else if (_M_key_compare(__k, _S_key(__position._M_node)))
        {
          // First, try before...
          iterator __before(__position._M_node);
          if (__position._M_node == _M_leftmost()) // begin()
            return _Res(_M_leftmost(), _M_leftmost());
          else if (_M_key_compare(_S_key((--__before)._M_node), __k))
            {
              if (!_S_right(__before._M_node))
                return _Res(_Base_ptr(), __before._M_node);
              else
                return _Res(__position._M_node, __position._M_node);
            }
          else
            return _M_get_insert_unique_pos(__k);
        }
      else if (_M_key_compare(_S_key(__position._M_node), __k))
        {
          // ... then try after.
          iterator __after(__position._M_node);
          if (__position._M_node == _M_rightmost())
            return _Res(_Base_ptr(), _M_rightmost());
          else if (_M_key_compare(__k, _S_key((++__after)._M_node)))
            {
              if (!_S_right(__position._M_node))
                return _Res(_Base_ptr(), __position._M_node);
              else
                return _Res(__after._M_node, __after._M_node);
            }
          else
            return _M_get_insert_unique_pos(__k);
        }
      else
        // Equivalent keys.
        return _Res(__position._M_node, _Base_ptr());
    }
 
#ifdef __glibcxx_associative_heterogeneous_insertion  // C++26
  template <typename _Key, typename _Val, typename _KeyOfValue,
            typename _Compare, typename _Alloc>
    template <typename _Kt>
      auto
      _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
      _M_get_insert_hint_unique_pos_tr(const_iterator __hint, const _Kt& __k)
      -> pair<_Base_ptr, _Base_ptr>
      {
        auto __node =__hint._M_node;
        if (__node == _M_end())
          {
            if (size() > 0 && _M_key_compare(_S_key(_M_rightmost()), __k))
              return { {}, _M_rightmost() };
            return _M_get_insert_unique_pos_tr(__k);
          }
        if (_M_key_compare(__k, _S_key(__node)))
          { // First, try before...
            if (__node == _M_leftmost()) // begin()
                return { _M_leftmost(), _M_leftmost() };
            iterator __before(__node);
            --__before;
            if (_M_key_compare(_S_key(__before._M_node), __k))
              {
                if (!_S_right(__before._M_node))
                  return { {}, __before._M_node }; // put right
                return { __node, __node }; // put left;
              }
            return _M_get_insert_unique_pos_tr(__k);
          }
        if (_M_key_compare(_S_key(__node), __k))
          { // ... then try after.
            if (__node == _M_rightmost())
              return { {}, _M_rightmost() };
            iterator __after(__node);
            ++__after;
            if (_M_key_compare(__k, _S_key(__after._M_node)))
              {
                if (!_S_right(__node))
                  return { {}, __node };
                return { __after._M_node, __after._M_node };
              }
            return _M_get_insert_unique_pos_tr(__k);
          }
        // Equal to __k; check if any more to the left.
        iterator __before(__node);
        if (__node == _M_leftmost() ||
              _M_key_compare(_S_key((--__before)._M_node), __k))
          { return { __node, {} }; }
        return _M_get_insert_unique_pos_tr(__k);
      }
#endif
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
#if __cplusplus >= 201103L
    template<typename _Arg, typename _NodeGen>
#else
    template<typename _NodeGen>
#endif
      typename _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::iterator
      _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
      _M_insert_unique_(const_iterator __position,
#if __cplusplus >= 201103L
                        _Arg&& __v,
#else
                        const _Val& __v,
#endif
                        _NodeGen& __node_gen)
    {
      pair<_Base_ptr, _Base_ptr> __res
        = _M_get_insert_hint_unique_pos(__position, _KeyOfValue()(__v));
 
      if (__res.second)
        return _M_insert_(__res.first, __res.second,
                          _GLIBCXX_FORWARD(_Arg, __v),
                          __node_gen);
      return iterator(__res.first);
    }
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
    pair<typename _Rb_tree<_Key, _Val, _KeyOfValue,
                           _Compare, _Alloc>::_Base_ptr,
         typename _Rb_tree<_Key, _Val, _KeyOfValue,
                           _Compare, _Alloc>::_Base_ptr>
    _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
    _M_get_insert_hint_equal_pos(const_iterator __position, const key_type& __k)
    {
      typedef pair<_Base_ptr, _Base_ptr> _Res;
 
      // end()
      if (__position._M_node == _M_end())
        {
          if (size() > 0
              && !_M_key_compare(__k, _S_key(_M_rightmost())))
            return _Res(_Base_ptr(), _M_rightmost());
          else
            return _M_get_insert_equal_pos(__k);
        }
      else if (!_M_key_compare(_S_key(__position._M_node), __k))
        {
          // First, try before...
          iterator __before(__position._M_node);
          if (__position._M_node == _M_leftmost()) // begin()
            return _Res(_M_leftmost(), _M_leftmost());
          else if (!_M_key_compare(__k, _S_key((--__before)._M_node)))
            {
              if (!_S_right(__before._M_node))
                return _Res(_Base_ptr(), __before._M_node);
              else
                return _Res(__position._M_node, __position._M_node);
            }
          else
            return _M_get_insert_equal_pos(__k);
        }
      else
        {
          // ... then try after.
          iterator __after(__position._M_node);
          if (__position._M_node == _M_rightmost())
            return _Res(_Base_ptr(), _M_rightmost());
          else if (!_M_key_compare(_S_key((++__after)._M_node), __k))
            {
              if (!_S_right(__position._M_node))
                return _Res(_Base_ptr(), __position._M_node);
              else
                return _Res(__after._M_node, __after._M_node);
            }
          else
            return _Res(_Base_ptr(), _Base_ptr());
        }
    }
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
#if __cplusplus >= 201103L
    template<typename _Arg, typename _NodeGen>
#else
    template<typename _NodeGen>
#endif
      typename _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::iterator
      _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
      _M_insert_equal_(const_iterator __position,
#if __cplusplus >= 201103L
                       _Arg&& __v,
#else
                       const _Val& __v,
#endif
                       _NodeGen& __node_gen)
      {
        pair<_Base_ptr, _Base_ptr> __res
          = _M_get_insert_hint_equal_pos(__position, _KeyOfValue()(__v));
 
        if (__res.second)
          return _M_insert_(__res.first, __res.second,
                            _GLIBCXX_FORWARD(_Arg, __v),
                            __node_gen);
 
        return _M_insert_equal_lower(_GLIBCXX_FORWARD(_Arg, __v));
      }
 
#if __cplusplus >= 201103L
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
    auto
    _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
    _M_insert_node(_Base_ptr __x, _Base_ptr __p, _Node_ptr __z)
    -> iterator
    {
      bool __insert_left = (__x || __p == _M_end()
                            || _M_key_compare(_S_key(__z), _S_key(__p)));
 
      _Base_ptr __base_z = __z->_M_base_ptr();
      _Node_traits::_S_insert_and_rebalance
        (__insert_left, __base_z, __p, this->_M_impl._M_header);
      ++_M_impl._M_node_count;
      return iterator(__base_z);
    }
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
    auto
    _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
    _M_insert_lower_node(_Base_ptr __p, _Node_ptr __z)
    -> iterator
    {
      bool __insert_left = (__p == _M_end()
                            || !_M_key_compare(_S_key(__p), _S_key(__z)));
 
      _Base_ptr __base_z = __z->_M_base_ptr();
      _Node_traits::_S_insert_and_rebalance
        (__insert_left, __base_z, __p, this->_M_impl._M_header);
      ++_M_impl._M_node_count;
      return iterator(__base_z);
    }
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
    auto
    _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
    _M_insert_equal_lower_node(_Node_ptr __z)
    -> iterator
    {
      _Base_ptr __x = _M_begin();
      _Base_ptr __y = _M_end();
      while (__x)
        {
          __y = __x;
          __x = !_M_key_compare(_S_key(__x), _S_key(__z)) ?
                _S_left(__x) : _S_right(__x);
        }
      return _M_insert_lower_node(__y, __z);
    }
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
    template<typename... _Args>
      auto
      _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
      _M_emplace_unique(_Args&&... __args)
      -> pair<iterator, bool>
      {
        _Auto_node __z(*this, std::forward<_Args>(__args)...);
        auto __res = _M_get_insert_unique_pos(__z._M_key());
        if (__res.second)
          return {__z._M_insert(__res), true};
        return {iterator(__res.first), false};
      }
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
    template<typename... _Args>
      auto
      _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
      _M_emplace_equal(_Args&&... __args)
      -> iterator
      {
        _Auto_node __z(*this, std::forward<_Args>(__args)...);
        auto __res = _M_get_insert_equal_pos(__z._M_key());
        return __z._M_insert(__res);
      }
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
    template<typename... _Args>
      auto
      _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
      _M_emplace_hint_unique(const_iterator __pos, _Args&&... __args)
      -> iterator
      {
        _Auto_node __z(*this, std::forward<_Args>(__args)...);
        auto __res = _M_get_insert_hint_unique_pos(__pos, __z._M_key());
        if (__res.second)
          return __z._M_insert(__res);
        return iterator(__res.first);
      }
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
    template<typename... _Args>
      auto
      _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
      _M_emplace_hint_equal(const_iterator __pos, _Args&&... __args)
      -> iterator
      {
        _Auto_node __z(*this, std::forward<_Args>(__args)...);
        auto __res = _M_get_insert_hint_equal_pos(__pos, __z._M_key());
        if (__res.second)
          return __z._M_insert(__res);
        return __z._M_insert_equal_lower();
      }
 
#ifdef __glibcxx_associative_heterogeneous_insertion  // C++26
  template <typename _Key, typename _Val, typename _KeyOfValue,
            typename _Compare, typename _Alloc>
    template <typename... _Args>
      auto
      _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
      _M_emplace_here(bool __place_left, _Base_ptr __node, _Args&&... __args)
      -> iterator
      {
        _Auto_node __z(*this, std::forward<_Args>(__args)...);
        _Base_ptr __base_z = __z._M_node->_M_base_ptr();
        _Node_traits::_S_insert_and_rebalance(
          __place_left, __base_z, __node, _M_impl._M_header);
        __z._M_node = nullptr;
        ++_M_impl._M_node_count;
        return iterator(__base_z);
      }
#endif
 
#endif  // >= C++11
 
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
    void
    _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
    _M_erase_aux(const_iterator __position)
    {
      _Base_ptr __y = _Node_traits::_S_rebalance_for_erase
        (__position._M_node, this->_M_impl._M_header);
      _M_drop_node(static_cast<_Node&>(*__y)._M_node_ptr());
      --_M_impl._M_node_count;
    }
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
    void
    _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
    _M_erase_aux(const_iterator __first, const_iterator __last)
    {
      if (__first == begin() && __last == end())
        clear();
      else
        while (__first != __last)
          _M_erase_aux(__first++);
    }
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
    typename _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::size_type
    _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
    erase(const _Key& __x)
    {
      pair<iterator, iterator> __p = equal_range(__x);
      const size_type __old_size = size();
      _M_erase_aux(__p.first, __p.second);
      return __old_size - size();
    }
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
    template <typename _Kt>
      typename _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::size_type
      _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
      _M_erase_tr(const _Kt& __x)
      {
        pair<iterator, iterator> __p = _M_equal_range_tr(__x);
        const size_type __old_size = size();
        _M_erase_aux(__p.first, __p.second);
        return __old_size - size();
      }
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
    typename _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::size_type
    _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
    _M_erase_unique(const _Key& __x)
    {
      iterator __it = find(__x);
      if (__it == end())
        return 0;
 
      _M_erase_aux(__it);
      return 1;
    }
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
    typename _Rb_tree<_Key, _Val, _KeyOfValue,
                      _Compare, _Alloc>::iterator
    _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
    find(const _Key& __k)
    {
      iterator __j(_M_lower_bound(_M_begin(), _M_end(), __k));
      return (__j == end()
              || _M_key_compare(__k, _S_key(__j._M_node))) ? end() : __j;
    }
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
    typename _Rb_tree<_Key, _Val, _KeyOfValue,
                      _Compare, _Alloc>::const_iterator
    _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
    find(const _Key& __k) const
    {
      const_iterator __j(_M_lower_bound(_M_begin(), _M_end(), __k));
      return (__j == end()
              || _M_key_compare(__k, _S_key(__j._M_node))) ? end() : __j;
    }
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
    typename _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::size_type
    _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::
    count(const _Key& __k) const
    {
      pair<const_iterator, const_iterator> __p = equal_range(__k);
      const size_type __n = std::distance(__p.first, __p.second);
      return __n;
    }
 
  _GLIBCXX_PURE unsigned int
  _Rb_tree_black_count(const _Rb_tree_node_base* __node,
                       const _Rb_tree_node_base* __root) throw ();
 
  template<typename _Key, typename _Val, typename _KeyOfValue,
           typename _Compare, typename _Alloc>
    bool
    _Rb_tree<_Key,_Val,_KeyOfValue,_Compare,_Alloc>::__rb_verify() const
    {
      if (_M_impl._M_node_count == 0 || begin() == end())
        return _M_impl._M_node_count == 0 && begin() == end()
               && this->_M_impl._M_header._M_left == _M_end()
               && this->_M_impl._M_header._M_right == _M_end();
 
      unsigned int __len = _Rb_tree_black_count(_M_leftmost(), _M_root());
      for (const_iterator __it = begin(); __it != end(); ++__it)
        {
          _Base_ptr __x = __it._M_node;
          _Base_ptr __L = _S_left(__x);
          _Base_ptr __R = _S_right(__x);
 
          if (__x->_M_color == _S_red)
            if ((__L && __L->_M_color == _S_red)
                || (__R && __R->_M_color == _S_red))
              return false;
 
          if (__L && _M_key_compare(_S_key(__x), _S_key(__L)))
            return false;
          if (__R && _M_key_compare(_S_key(__R), _S_key(__x)))
            return false;
 
          if (!__L && !__R && _Rb_tree_black_count(__x, _M_root()) != __len)
            return false;
        }
 
      if (_M_leftmost() != _Node_base::_S_minimum(_M_root()))
        return false;
      if (_M_rightmost() != _Node_base::_S_maximum(_M_root()))
        return false;
      return true;
    }
 
#ifdef __glibcxx_node_extract // >= C++17
  // Allow access to internals of compatible _Rb_tree specializations.
  template<typename _Key, typename _Val, typename _Sel, typename _Cmp1,
           typename _Alloc, typename _Cmp2>
    struct _Rb_tree_merge_helper<_Rb_tree<_Key, _Val, _Sel, _Cmp1, _Alloc>,
                                 _Cmp2>
    {
    private:
      friend class _Rb_tree<_Key, _Val, _Sel, _Cmp1, _Alloc>;
 
      static auto&
      _S_get_impl(_Rb_tree<_Key, _Val, _Sel, _Cmp2, _Alloc>& __tree)
      { return __tree._M_impl; }
    };
#endif // C++17
 
#ifdef __glibcxx_associative_heterogeneous_erasure // C++ >= 23
template <typename _Kt, typename _Container>
  concept __heterogeneous_tree_key =
    __transparent_comparator<typename _Container::key_compare> &&
    __heterogeneous_key<_Kt, _Container>;
#endif
 
_GLIBCXX_END_NAMESPACE_VERSION
} // namespace
 
#endif