STL源码剖析——RB-Tree(红黑树)

叁歲伎倆 2022-08-12 11:30 157阅读 0赞

## 前言 ##

有关红黑树的知识在前面博文有介绍，读者可自行往前面博文《[红黑树RB-Tree][RB-Tree]》，《[二叉查找树][Link 1]》阅读。本文介绍的RB-Tree(红黑树)是来自源码SGI STL中的文件<stl\_tree.h>。这里采用了header技巧，header指向根节点的指针，与根节点互为对方的父节点。他们之间的结构如下图所示：

![SouthEast][]

## RB-Tree(红黑树)源码剖析    ##

#ifndef __SGI_STL_INTERNAL_TREE_H
    #define __SGI_STL_INTERNAL_TREE_H
    
    /*
    
    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.
    
    */
    //下面这段话是侯捷老师对上面说明的翻译.
    /*
    
    这章讲解Red-black tree（红黑树）class,用来当做SLT关系式容器（如set,multiset,map, 
    multimap）.里面所用的insertion和deletion方法以Cormen, Leiserson 和 Riveset所著的 
    《Introduction to Algorithms》一书为基础,但是有以下两点不同: 
     
    (1)header不仅指向root,也指向红黑树的最左节点,以便用常数时间实现begin(),并且也指向红黑树的最右边节点,以便 
    set相关泛型算法（如set_union等等）可以有线性时间表现.
    
    (2)当一个即将被删除的节点有两个孩子节点时,它的successor（后继）node is relinked into its place, ranther than copied, 
    如此一来唯一失效的（invalidated）的迭代器就只是那些referring to the deleted node.  
    */
    
    /*	
    	SGI STL中的RB-Tree实现机制有一定的技巧,定义了一个指向根节点的节点指针header,
    并且,header和根节点root互为对方的父节点,header的左子节点指向RB-Tree的最小节点,
    header的右子节点指向RB-Tree的最大节点.
    */
    
    /*
    RB-Tree是一棵二叉查找树,并且具备有以下性质:
    红黑树的性质：
    	(1)每个节点或是红色的,或是黑色的.
    	(2)根节点是黑色的.
    	(3)每个叶节点（NULL）是黑色的.
    	(4)如果一个节点是红色的，则它的两个孩子节点都是黑色的.
    	(5)对每个节点，从该节点到其所有后代叶节点的简单路径上，均包含相同数目的黑色节点.
    */
    #include <stl_algobase.h>
    #include <stl_alloc.h>
    #include <stl_construct.h>
    #include <stl_function.h>
    
    __STL_BEGIN_NAMESPACE 
    
    #if defined(__sgi) && !defined(__GNUC__) && (_MIPS_SIM != _MIPS_SIM_ABI32)
    #pragma set woff 1375
    #endif
    
    typedef bool _Rb_tree_Color_type;//节点颜色类型，非红即黑
    const _Rb_tree_Color_type _S_rb_tree_red = false;//红色为0
    const _Rb_tree_Color_type _S_rb_tree_black = true;//黑色为1
    
    //RB-Tree节点基本结构
    struct _Rb_tree_node_base
    {
      typedef _Rb_tree_Color_type _Color_type;//节点颜色类型
      typedef _Rb_tree_node_base* _Base_ptr;//基本节点指针
    
      _Color_type _M_color; //节点颜色
      _Base_ptr _M_parent;//指向父节点
      _Base_ptr _M_left;//指向左孩子节点
      _Base_ptr _M_right;//指向右孩子节点
    
      //RB-Tree最小节点,即最左节点
      static _Base_ptr _S_minimum(_Base_ptr __x)
      {
        
    	while (__x->_M_left != 0) __x = __x->_M_left;
        return __x;
      }
    
      //RB-Tree最大节点,即最右节点
      static _Base_ptr _S_maximum(_Base_ptr __x)
      {
        //一直往RB-Tree的右边走,最右的节点即是最大节点
    	//这是二叉查找树的性质
    	while (__x->_M_right != 0) __x = __x->_M_right;
        return __x;
      }
    };
    
    //RB-Tree节点结构
    //继承节点基本结构_Rb_tree_node_base的节点信息
    template <class _Value>
    struct _Rb_tree_node : public _Rb_tree_node_base
    {
      typedef _Rb_tree_node<_Value>* _Link_type;//节点指针,指向数据节点
      _Value _M_value_field;//节点数据域,即关键字
    };
    
    
    //RB-Tree的迭代器iterator基本结构
    //iterator迭代器的类型为双向迭代器bidirectional_iterator
    struct _Rb_tree_base_iterator
    {
      typedef _Rb_tree_node_base::_Base_ptr _Base_ptr;//节点指针
      typedef bidirectional_iterator_tag iterator_category;//迭代器类型
      typedef ptrdiff_t difference_type;
      _Base_ptr _M_node;//节点指针,连接容器
    
    	/* 
        重载运算符++和--;目的是找到前驱和后继节点. 
    	*/  
      //下面只是为了实现oprerator++的,其他地方不会调用.  
      //RB-Tree的后继节点
      void _M_increment()
      {
        //the right subtree of node x is not empty
    	  //【情况1】:存在右子树，则找出右子树的最小节点
    	if (_M_node->_M_right != 0) {//如果有右子树
          _M_node = _M_node->_M_right;//向右边走
          while (_M_node->_M_left != 0)//往右子树中的左边一直走到底
            _M_node = _M_node->_M_left;//最左节点就是后继结点
        }
    	//the right subtree of node x is empty,and the node of x has a successor node y 
    	//没有右子树，但是RB-Tree中节点node存在后继结点
        else {
          _Base_ptr __y = _M_node->_M_parent;//沿其父节点上溯
          while (_M_node == __y->_M_right) { //【情况2】:若节点是其父节点的右孩子,则上溯
            _M_node = __y;					//一直上溯,直到“某节点不是其父节点的右孩子”为止
            __y = __y->_M_parent;
          }
    	  /* 
            若此时的右子节点不等于此时的父节点,此时的父节点即为解答,否则此时的node为解答.
    		因为SGI STL中的RB-Tree加入的header节点，所以需考虑特殊情况;
            这样做是为了应付一种特殊情况：我们欲寻找根节点的下一个节点，而恰巧根节点无右孩子。 
            当然，以上特殊做法必须配合RB-tree根节点与特殊header之间的特殊关系
          */
    	  //以下情况3和情况4是针对header节点的使用,因为根节点和header节点是互为父节点
          if (_M_node->_M_right != __y)//【情况3】:若此时的右子节点不等于此时的父节点
            _M_node = __y;//此时的父节点即为解答
    						//【情况4】：否则此时的node为解答 
        }
      }
    
      //下面只是为了实现oprerator--的,其他地方不会调用.  
      //RB-Tree的前驱节点
      void _M_decrement()
      {
        if (_M_node->_M_color == _S_rb_tree_red &&// 如果是红节点，且
            _M_node->_M_parent->_M_parent == _M_node)// 父节点的父节点等于自己
          _M_node = _M_node->_M_right;			 //【情况1】：右子节点即为解答。
    	/* 
          以上情况发生于node为header时（亦即node为end()时）。注意，header之右孩子即 
          mostright，指向整棵树的max节点。 
          */ 
        else if (_M_node->_M_left != 0) {//若有左孩子节点。【情况2】:左子树的最大值即为前驱节点
          _Base_ptr __y = _M_node->_M_left;//向左边走,即令y指向左孩子
          while (__y->_M_right != 0)//y存在右孩子,
            __y = __y->_M_right;//一直往右走到底
          _M_node = __y;//最后即为解答
        }
        else {//即非根节点,且没有左孩子节点,【情况3】
          _Base_ptr __y = _M_node->_M_parent;//找出父节点
          while (_M_node == __y->_M_left) {//node节点是其父节点的左孩子
            _M_node = __y;//一直交替上溯
            __y = __y->_M_parent;//直到不为左孩子结点
          }
          _M_node = __y;//此时父节点即为解答
        }
      }
    };
    
    //RB-Tree的迭代器iterator结构
    //继承基类迭代器Rb_tree_base_iterator
    template <class _Value, class _Ref, class _Ptr>
    struct _Rb_tree_iterator : public _Rb_tree_base_iterator
    {
      //迭代器的内嵌类型
      typedef _Value value_type;
      typedef _Ref reference;
      typedef _Ptr pointer;
      typedef _Rb_tree_iterator<_Value, _Value&, _Value*>             
        iterator;
      typedef _Rb_tree_iterator<_Value, const _Value&, const _Value*> 
        const_iterator;
      typedef _Rb_tree_iterator<_Value, _Ref, _Ptr>                   
        _Self;
      typedef _Rb_tree_node<_Value>* _Link_type;//节点指针
    
      //构造函数
      _Rb_tree_iterator() {}
      _Rb_tree_iterator(_Link_type __x) { _M_node = __x; }
      _Rb_tree_iterator(const iterator& __it) { _M_node = __it._M_node; }
    
      /*
      以下是操作符重载
      */
      reference operator*() const { return _Link_type(_M_node)->_M_value_field; }
    #ifndef __SGI_STL_NO_ARROW_OPERATOR
      pointer operator->() const { return &(operator*()); }
    #endif /* __SGI_STL_NO_ARROW_OPERATOR */
    
      //前缀operator++找出后继节点,调用基类的_M_increment()
      _Self& operator++() { _M_increment(); return *this; }
      //后缀operator++找出后继节点,调用基类的_M_increment()
      _Self operator++(int) {
        _Self __tmp = *this;
        _M_increment();
        return __tmp;
      }
        
      //前缀operator--找出前驱节点,调用基类的_M_decrement()
      _Self& operator--() { _M_decrement(); return *this; }
      //后缀operator--找出前驱节点,调用基类的_M_decrement()
      _Self operator--(int) {
        _Self __tmp = *this;
        _M_decrement();
        return __tmp;
      }
    };
    
    //两个迭代器相等，意味着指向RB-Tree的同一个节点
    inline bool operator==(const _Rb_tree_base_iterator& __x,
                           const _Rb_tree_base_iterator& __y) {
      return __x._M_node == __y._M_node;
    }
    
    inline bool operator!=(const _Rb_tree_base_iterator& __x,
                           const _Rb_tree_base_iterator& __y) {
      return __x._M_node != __y._M_node;
      //两个迭代器不相等，意味着所指向的节点不同
    }
    
    #ifndef __STL_CLASS_PARTIAL_SPECIALIZATION
    
    inline bidirectional_iterator_tag
    iterator_category(const _Rb_tree_base_iterator&) {
      return bidirectional_iterator_tag();
    }
    
    inline _Rb_tree_base_iterator::difference_type*
    distance_type(const _Rb_tree_base_iterator&) {
      return (_Rb_tree_base_iterator::difference_type*) 0;
    }
    
    template <class _Value, class _Ref, class _Ptr>
    inline _Value* value_type(const _Rb_tree_iterator<_Value, _Ref, _Ptr>&) {
      return (_Value*) 0;
    }
    
    #endif /* __STL_CLASS_PARTIAL_SPECIALIZATION */
    
    // 以下都是全域函式：__rb_tree_rotate_left(), __rb_tree_rotate_right(),  
    // __rb_tree_rebalance(), __rb_tree_rebalance_for_erase()  
      
    //新节点必须为红色节点。如果安插处的父节点为红色，就违反了红黑色规则
    //此时要旋转和改变颜色 
    
    //左旋转
    //节点x为左旋转点
    inline void 
    _Rb_tree_rotate_left(_Rb_tree_node_base* __x, _Rb_tree_node_base*& __root)
    {
      _Rb_tree_node_base* __y = __x->_M_right;//获取左旋转节点x的右孩子y
      __x->_M_right = __y->_M_left;//把y节点的左孩子作为旋转节点x的右孩子
      if (__y->_M_left !=0)
        __y->_M_left->_M_parent = __x;//更新节点y左孩子父节点指针，指向新的父节点x
      __y->_M_parent = __x->_M_parent;//y节点替换x节点的位置
    
      //令y完全顶替x的地位（必须将x对其父节点的关系完全接收过来）
      if (__x == __root)//若原始位置节点x是根节点
        __root = __y;//则y为新的根节点
      //否则，若x节点是其父节点的左孩子
      else if (__x == __x->_M_parent->_M_left)
        __x->_M_parent->_M_left = __y;//则更新节点y为原始x父节点的左孩子
      else//若x节点是其父节点的右孩子
        __x->_M_parent->_M_right = __y;//则更新节点y为原始x父节点的右孩子
      __y->_M_left = __x;//旋转后旋转节点x作为节点y的左孩子
      __x->_M_parent = __y;//更新x节点的父节点指针
    }
    
    //右旋转
    //节点x为右旋转点
    inline void 
    _Rb_tree_rotate_right(_Rb_tree_node_base* __x, _Rb_tree_node_base*& __root)
    {
      _Rb_tree_node_base* __y = __x->_M_left;//获取右旋转节点x的左孩子y
      __x->_M_left = __y->_M_right;//把y节点的右孩子作为旋转节点x的左孩子
      if (__y->_M_right != 0)
        __y->_M_right->_M_parent = __x;//更新节点y右孩子父节点指针，指向新的父节点x
      __y->_M_parent = __x->_M_parent;//y节点替换x节点的位置
    
      //令y完全顶替x的地位（必须将x对其父节点的关系完全接收过来）
      if (__x == __root)//若原始位置节点x是根节点
        __root = __y;//则y为新的根节点
      //否则，若x节点是其父节点的右孩子
      else if (__x == __x->_M_parent->_M_right)
        __x->_M_parent->_M_right = __y;//则更新节点y为原始x父节点的右孩子
      else//若x节点是其父节点的左孩子
        __x->_M_parent->_M_left = __y;//则更新节点y为原始x父节点的左孩子
      __y->_M_right = __x;//旋转后旋转节点x作为节点y的右孩子
      __x->_M_parent = __y;//更新x节点的父节点指针
    }
    
    //重新令RB-tree平衡（改变颜色和旋转）
    //参数一为新增节点x，参数二为root节点
    inline void 
    _Rb_tree_rebalance(_Rb_tree_node_base* __x, _Rb_tree_node_base*& __root)
    {
      __x->_M_color = _S_rb_tree_red;//新插入的节点必须为红色,这样不会违反性质5.
      //若新插入节点不是为RB-Tree的根节点，且其父节点color属性也是红色,即违反了性质4.
      //则进入while循环.
      //此时根据节点x的父节点x->parent是其祖父节点x->parent->parent的左孩子还是右孩子进行讨论,
      //但是左右孩子之间是对称的，所以思想是类似的.
      while (__x != __root && __x->_M_parent->_M_color == _S_rb_tree_red) {
    	//case1：节点x的父节点x->parent是其祖父节点x->parent->parent的左孩子
        if (__x->_M_parent == __x->_M_parent->_M_parent->_M_left) {
    		//节点y为x节点的叔叔节点,即是节点x父节点x->parent的兄弟
          _Rb_tree_node_base* __y = __x->_M_parent->_M_parent->_M_right;
          if (__y && __y->_M_color == _S_rb_tree_red) {//情况1：若其叔叔节点y存在,且为红色
    		  /*
    		  此时x->parent和y都是红色的，解决办法是将x的父节点x->parent和叔叔结点y都着为黑色，
    		  而将x的祖父结点x->parent->parent着为红色，
    		  然后从祖父结点x->parent->parent继续向上判断是否破坏红黑树的性质。
    		  */
            __x->_M_parent->_M_color = _S_rb_tree_black;//将其父节点x->parent改变成黑色
            __y->_M_color = _S_rb_tree_black;//将其叔叔节点y改变成黑色
            __x->_M_parent->_M_parent->_M_color = _S_rb_tree_red;//将其祖父节点变成红色
    		//把祖父节点作为当前节点,一直上溯,继续判断是否破坏RB-Tree性质.
            __x = __x->_M_parent->_M_parent;
          }
          else {//若无叔叔节点或者其叔叔节点y为黑色
    		  /*
    		  情况2：x的叔叔节点y是黑色且x是一个右孩子
    		  情况3：x的叔叔节点y是黑色且x是一个左孩子
    
    		 情况2和情况3中y都是黑色的，通过x是parent[x]的左孩子还是右孩子进行区分的。
    		 情况2中x是右孩子，可以在parent[x]结点将情况2通过左旋转为情况3，使得x变为左孩子。
    		 无论是间接还是直接的通过情况2进入到情况3，x的叔叔y总是黑色的。
    		 在情况3中，将parent[x]着为黑色，parent[parent[x]]着为红色，然后从parent[parent[x]]处进行一次右旋转。
    		 情况2、3修正了对性质4的违反，修正过程不会导致其他的红黑性质被破坏。
    		  */
            if (__x == __x->_M_parent->_M_right) {//若节点x为其父节点x->parent的右孩子
    			//则以其父节点作为旋转节点
    			//进行一次左旋转
              __x = __x->_M_parent;
              _Rb_tree_rotate_left(__x, __root);
    		  //旋转之后,节点x变成其父节点的左孩子
            }
            __x->_M_parent->_M_color = _S_rb_tree_black;//改变其父节点x->parent颜色
            __x->_M_parent->_M_parent->_M_color = _S_rb_tree_red;//改变其祖父节点x->parent->parent颜色
            _Rb_tree_rotate_right(__x->_M_parent->_M_parent, __root);//对其祖父节点进行一次右旋转
          }
        }
    	//case2：节点x的父节点x->parent是其祖父节点x->parent->parent的右孩子
    	//这种情况是跟上面的情况(父节点为其祖父节点的左孩子)是对称的.
        else {
    		//节点y为x节点的叔叔节点,即是节点x父节点x->parent的兄弟
          _Rb_tree_node_base* __y = __x->_M_parent->_M_parent->_M_left;
          if (__y && __y->_M_color == _S_rb_tree_red) {//若叔叔节点存在,且为红色
            __x->_M_parent->_M_color = _S_rb_tree_black;//改变父节点颜色
            __y->_M_color = _S_rb_tree_black;//改变叔叔节点颜色
            __x->_M_parent->_M_parent->_M_color = _S_rb_tree_red;//改变祖父节点颜色
            __x = __x->_M_parent->_M_parent;//上溯祖父节点,判断是否违背RB-Tree的性质
          }
          else {//若叔叔节点不存在或叔叔节点为黑色
            if (__x == __x->_M_parent->_M_left) {//新节点x为其父节点的左孩子
    			//对其父节点进行一次右旋转
              __x = __x->_M_parent;
              _Rb_tree_rotate_right(__x, __root);
            }
            __x->_M_parent->_M_color = _S_rb_tree_black;//改变父节点颜色
            __x->_M_parent->_M_parent->_M_color = _S_rb_tree_red;//改变祖父节点颜色
            _Rb_tree_rotate_left(__x->_M_parent->_M_parent, __root);//进行一次左旋转
          }
        }
      }
      //若新插入节点为根节点,则违反性质2
      //只需将其重新赋值为黑色即可
      __root->_M_color = _S_rb_tree_black;
    }
    
    //删除节点
    inline _Rb_tree_node_base*
    _Rb_tree_rebalance_for_erase(_Rb_tree_node_base* __z,
                                 _Rb_tree_node_base*& __root,
                                 _Rb_tree_node_base*& __leftmost,
                                 _Rb_tree_node_base*& __rightmost)
    {
      _Rb_tree_node_base* __y = __z;
      _Rb_tree_node_base* __x = 0;
      _Rb_tree_node_base* __x_parent = 0;
      if (__y->_M_left == 0)     // __z has at most one non-null child. y == z.
        __x = __y->_M_right;     // __x might be null.
      else
        if (__y->_M_right == 0)  // __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 != 0)
            __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 == 0)        // __z->_M_left must be null also
            __leftmost = __z->_M_parent;
        // makes __leftmost == _M_header if __z == __root
          else
            __leftmost = _Rb_tree_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 = _Rb_tree_node_base::_S_maximum(__x);
      }
      if (__y->_M_color != _S_rb_tree_red) { 
        while (__x != __root && (__x == 0 || __x->_M_color == _S_rb_tree_black))
          if (__x == __x_parent->_M_left) {
            _Rb_tree_node_base* __w = __x_parent->_M_right;
            if (__w->_M_color == _S_rb_tree_red) {
              __w->_M_color = _S_rb_tree_black;
              __x_parent->_M_color = _S_rb_tree_red;
              _Rb_tree_rotate_left(__x_parent, __root);
              __w = __x_parent->_M_right;
            }
            if ((__w->_M_left == 0 || 
                 __w->_M_left->_M_color == _S_rb_tree_black) &&
                (__w->_M_right == 0 || 
                 __w->_M_right->_M_color == _S_rb_tree_black)) {
              __w->_M_color = _S_rb_tree_red;
              __x = __x_parent;
              __x_parent = __x_parent->_M_parent;
            } else {
              if (__w->_M_right == 0 || 
                  __w->_M_right->_M_color == _S_rb_tree_black) {
                if (__w->_M_left) __w->_M_left->_M_color = _S_rb_tree_black;
                __w->_M_color = _S_rb_tree_red;
                _Rb_tree_rotate_right(__w, __root);
                __w = __x_parent->_M_right;
              }
              __w->_M_color = __x_parent->_M_color;
              __x_parent->_M_color = _S_rb_tree_black;
              if (__w->_M_right) __w->_M_right->_M_color = _S_rb_tree_black;
              _Rb_tree_rotate_left(__x_parent, __root);
              break;
            }
          } else {                  // same as above, with _M_right <-> _M_left.
            _Rb_tree_node_base* __w = __x_parent->_M_left;
            if (__w->_M_color == _S_rb_tree_red) {
              __w->_M_color = _S_rb_tree_black;
              __x_parent->_M_color = _S_rb_tree_red;
              _Rb_tree_rotate_right(__x_parent, __root);
              __w = __x_parent->_M_left;
            }
            if ((__w->_M_right == 0 || 
                 __w->_M_right->_M_color == _S_rb_tree_black) &&
                (__w->_M_left == 0 || 
                 __w->_M_left->_M_color == _S_rb_tree_black)) {
              __w->_M_color = _S_rb_tree_red;
              __x = __x_parent;
              __x_parent = __x_parent->_M_parent;
            } else {
              if (__w->_M_left == 0 || 
                  __w->_M_left->_M_color == _S_rb_tree_black) {
                if (__w->_M_right) __w->_M_right->_M_color = _S_rb_tree_black;
                __w->_M_color = _S_rb_tree_red;
                _Rb_tree_rotate_left(__w, __root);
                __w = __x_parent->_M_left;
              }
              __w->_M_color = __x_parent->_M_color;
              __x_parent->_M_color = _S_rb_tree_black;
              if (__w->_M_left) __w->_M_left->_M_color = _S_rb_tree_black;
              _Rb_tree_rotate_right(__x_parent, __root);
              break;
            }
          }
        if (__x) __x->_M_color = _S_rb_tree_black;
      }
      return __y;
    }
    
    // Base class to encapsulate the differences between old SGI-style
    // allocators and standard-conforming allocators.  In order to avoid
    // having an empty base class, we arbitrarily move one of rb_tree's
    // data members into the base class.
    
    //以下是对内存分配的管理
    #ifdef __STL_USE_STD_ALLOCATORS
    
    // _Base for general standard-conforming allocators.
    template <class _Tp, class _Alloc, bool _S_instanceless>
    class _Rb_tree_alloc_base {
    public:
      typedef typename _Alloc_traits<_Tp, _Alloc>::allocator_type allocator_type;
      allocator_type get_allocator() const { return _M_node_allocator; }//空间配置器的类型
    
      _Rb_tree_alloc_base(const allocator_type& __a)
        : _M_node_allocator(__a), _M_header(0) {}
    
    protected:
      typename _Alloc_traits<_Rb_tree_node<_Tp>, _Alloc>::allocator_type
               _M_node_allocator;
      _Rb_tree_node<_Tp>* _M_header;//定义头指针，指向Rb_tree的根节点
    
      _Rb_tree_node<_Tp>* _M_get_node() //分配一个节点空间
        { return _M_node_allocator.allocate(1); }
      void _M_put_node(_Rb_tree_node<_Tp>* __p) //释放一个节点空间
        { _M_node_allocator.deallocate(__p, 1); }
    };
    
    // Specialization for instanceless allocators.
    template <class _Tp, class _Alloc>
    class _Rb_tree_alloc_base<_Tp, _Alloc, true> {
    public:
      typedef typename _Alloc_traits<_Tp, _Alloc>::allocator_type allocator_type;
      allocator_type get_allocator() const { return allocator_type(); }
    
      _Rb_tree_alloc_base(const allocator_type&) : _M_header(0) {}
    
    protected:
      _Rb_tree_node<_Tp>* _M_header;
    
      typedef typename _Alloc_traits<_Rb_tree_node<_Tp>, _Alloc>::_Alloc_type
              _Alloc_type;
    
      _Rb_tree_node<_Tp>* _M_get_node()
        { return _Alloc_type::allocate(1); }
      void _M_put_node(_Rb_tree_node<_Tp>* __p)
        { _Alloc_type::deallocate(__p, 1); }
    };
    
    //RB-Tree基本结构,即基类,继承_Rb_tree_alloc_base
    template <class _Tp, class _Alloc>
    struct _Rb_tree_base
      : public _Rb_tree_alloc_base<_Tp, _Alloc,
                                   _Alloc_traits<_Tp, _Alloc>::_S_instanceless>
    {
      typedef _Rb_tree_alloc_base<_Tp, _Alloc,
                                  _Alloc_traits<_Tp, _Alloc>::_S_instanceless>
              _Base;
      typedef typename _Base::allocator_type allocator_type;
    
      _Rb_tree_base(const allocator_type& __a) 
        : _Base(__a) { _M_header = _M_get_node(); }
      ~_Rb_tree_base() { _M_put_node(_M_header); }
    
    };
    
    #else /* __STL_USE_STD_ALLOCATORS */
    
    //RB-Tree基本结构,即基类,没有继承_Rb_tree_alloc_base
    template <class _Tp, class _Alloc>
    struct _Rb_tree_base
    {
      typedef _Alloc allocator_type;
      allocator_type get_allocator() const { return allocator_type(); }
    
      _Rb_tree_base(const allocator_type&) 
        : _M_header(0) { _M_header = _M_get_node(); }
      ~_Rb_tree_base() { _M_put_node(_M_header); }
    
    protected:
      _Rb_tree_node<_Tp>* _M_header;//定义头指针节点，指向根节点
    
      typedef simple_alloc<_Rb_tree_node<_Tp>, _Alloc> _Alloc_type;
    
      _Rb_tree_node<_Tp>* _M_get_node()
        { return _Alloc_type::allocate(1); }
      void _M_put_node(_Rb_tree_node<_Tp>* __p)
        { _Alloc_type::deallocate(__p, 1); }
    };
    
    #endif /* __STL_USE_STD_ALLOCATORS */
    
    //RB-Tree类的定义,继承基类_Rb_tree_base
    template <class _Key, class _Value, class _KeyOfValue, class _Compare,
              class _Alloc = __STL_DEFAULT_ALLOCATOR(_Value) >
    class _Rb_tree : protected _Rb_tree_base<_Value, _Alloc> {
      typedef _Rb_tree_base<_Value, _Alloc> _Base;
    protected:
      typedef _Rb_tree_node_base* _Base_ptr;
      typedef _Rb_tree_node<_Value> _Rb_tree_node;
      typedef _Rb_tree_Color_type _Color_type;
    public:
      typedef _Key key_type;
      typedef _Value value_type;
      typedef value_type* pointer;
      typedef const value_type* const_pointer;
      typedef value_type& reference;
      typedef const value_type& const_reference;
      typedef _Rb_tree_node* _Link_type;
      typedef size_t size_type;
      typedef ptrdiff_t difference_type;
    
      typedef typename _Base::allocator_type allocator_type;
      allocator_type get_allocator() const { return _Base::get_allocator(); }
    
    protected:
    #ifdef __STL_USE_NAMESPACES
      using _Base::_M_get_node;
      using _Base::_M_put_node;
      using _Base::_M_header;//这里是指向根节点的节点指针
    #endif /* __STL_USE_NAMESPACES */
    
    protected:
    
    //创建节点并对其初始化为x  
    _Link_type _M_create_node(const value_type& __x)
      {
        _Link_type __tmp = _M_get_node();//分配一个节点空间
        __STL_TRY {
          construct(&__tmp->_M_value_field, __x);//构造对象
        }
        __STL_UNWIND(_M_put_node(__tmp));
        return __tmp;
      }
    
    //复制节点的值和颜色
      _Link_type _M_clone_node(_Link_type __x)
      {
        _Link_type __tmp = _M_create_node(__x->_M_value_field);
        __tmp->_M_color = __x->_M_color;
        __tmp->_M_left = 0;
        __tmp->_M_right = 0;
        return __tmp;
      }
    
      //释放节点
      void destroy_node(_Link_type __p)
      {
        destroy(&__p->_M_value_field);//析构对象
        _M_put_node(__p);//释放节点空间
      }
    
    protected:
      size_type _M_node_count; // keeps track of size of tree
      _Compare _M_key_compare;	//节点键值比较准则
    
      //下面三个函数是用来获取header的成员
      _Link_type& _M_root() const 
        { return (_Link_type&) _M_header->_M_parent; }
      _Link_type& _M_leftmost() const 
        { return (_Link_type&) _M_header->_M_left; }
      _Link_type& _M_rightmost() const 
        { return (_Link_type&) _M_header->_M_right; }
    
      //下面六个函数获取节点x的成员
      static _Link_type& _S_left(_Link_type __x)
        { return (_Link_type&)(__x->_M_left); }
      static _Link_type& _S_right(_Link_type __x)
        { return (_Link_type&)(__x->_M_right); }
      static _Link_type& _S_parent(_Link_type __x)
        { return (_Link_type&)(__x->_M_parent); }
      static reference _S_value(_Link_type __x)
        { return __x->_M_value_field; }
      static const _Key& _S_key(_Link_type __x)
        { return _KeyOfValue()(_S_value(__x)); }
      static _Color_type& _S_color(_Link_type __x)
        { return (_Color_type&)(__x->_M_color); }
    
      //跟上面六个函数功能相同，不同的是参数类型不同，一个是基类指针，一个是派生类指针  
      static _Link_type& _S_left(_Base_ptr __x)
        { return (_Link_type&)(__x->_M_left); }
      static _Link_type& _S_right(_Base_ptr __x)
        { return (_Link_type&)(__x->_M_right); }
      static _Link_type& _S_parent(_Base_ptr __x)
        { return (_Link_type&)(__x->_M_parent); }
      static reference _S_value(_Base_ptr __x)
        { return ((_Link_type)__x)->_M_value_field; }
      static const _Key& _S_key(_Base_ptr __x)
        { return _KeyOfValue()(_S_value(_Link_type(__x)));} 
      static _Color_type& _S_color(_Base_ptr __x)
        { return (_Color_type&)(_Link_type(__x)->_M_color); }
    
      //RB-Tree的极小值
      static _Link_type _S_minimum(_Link_type __x) 
        { return (_Link_type)  _Rb_tree_node_base::_S_minimum(__x); }
    
       //RB-Tree的极大值
      static _Link_type _S_maximum(_Link_type __x)
        { return (_Link_type) _Rb_tree_node_base::_S_maximum(__x); }
    
    public:
    	//迭代器
      typedef _Rb_tree_iterator<value_type, reference, pointer> iterator;
      typedef _Rb_tree_iterator<value_type, const_reference, const_pointer> 
              const_iterator;
    
    #ifdef __STL_CLASS_PARTIAL_SPECIALIZATION
      typedef reverse_iterator<const_iterator> const_reverse_iterator;
      typedef reverse_iterator<iterator> reverse_iterator;
    #else /* __STL_CLASS_PARTIAL_SPECIALIZATION */
      typedef reverse_bidirectional_iterator<iterator, value_type, reference,
                                             difference_type>
              reverse_iterator; 
      typedef reverse_bidirectional_iterator<const_iterator, value_type,
                                             const_reference, difference_type>
              const_reverse_iterator;
    #endif /* __STL_CLASS_PARTIAL_SPECIALIZATION */ 
    
    private:
    	//类的私有成员函数,在后面定义
      iterator _M_insert(_Base_ptr __x, _Base_ptr __y, const value_type& __v);
      _Link_type _M_copy(_Link_type __x, _Link_type __p);
      void _M_erase(_Link_type __x);
    
    public:
                                    // allocation/deallocation
      _Rb_tree()
        : _Base(allocator_type()), _M_node_count(0), _M_key_compare()
        { _M_empty_initialize(); }
    
      _Rb_tree(const _Compare& __comp)
        : _Base(allocator_type()), _M_node_count(0), _M_key_compare(__comp) 
        { _M_empty_initialize(); }
    
      _Rb_tree(const _Compare& __comp, const allocator_type& __a)
        : _Base(__a), _M_node_count(0), _M_key_compare(__comp) 
        { _M_empty_initialize(); }
    
      _Rb_tree(const _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>& __x) 
        : _Base(__x.get_allocator()),
          _M_node_count(0), _M_key_compare(__x._M_key_compare)
      { 
        if (__x._M_root() == 0)
          _M_empty_initialize();
        else {
          _S_color(_M_header) = _S_rb_tree_red;
          _M_root() = _M_copy(__x._M_root(), _M_header);
          _M_leftmost() = _S_minimum(_M_root());
          _M_rightmost() = _S_maximum(_M_root());
        }
        _M_node_count = __x._M_node_count;
      }
      ~_Rb_tree() { clear(); }
      _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>& 
      operator=(const _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>& __x);
    
    private:
    	//初始化header
      void _M_empty_initialize() {
        _S_color(_M_header) = _S_rb_tree_red; // used to distinguish header from 
                                              // __root, in iterator.operator++
        _M_root() = 0;
        _M_leftmost() = _M_header;
        _M_rightmost() = _M_header;
      }
    
    public:    
                                    // accessors:
      _Compare key_comp() const { return _M_key_compare; }
      iterator begin() { return _M_leftmost(); }//RB-Tree的起始迭代器为最小节点
      const_iterator begin() const { return _M_leftmost(); }
      iterator end() { return _M_header; }//RB-Tree的结束迭代器为header
      const_iterator end() const { return _M_header; }
      reverse_iterator rbegin() { return reverse_iterator(end()); }
      const_reverse_iterator rbegin() const { 
        return const_reverse_iterator(end()); 
      }
      reverse_iterator rend() { return reverse_iterator(begin()); }
      const_reverse_iterator rend() const { 
        return const_reverse_iterator(begin());
      } 
      //RB-Tree是否为空
      bool empty() const { return _M_node_count == 0; }
      //RB-Tree节点数
      size_type size() const { return _M_node_count; }
      size_type max_size() const { return size_type(-1); }
    
      //交换两棵RB-Tree的内容
      //RB-tree只有三个表现成员，所以两棵RB-Tree交换内容时，只需互换这3个成员 
      void swap(_Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>& __t) {
        __STD::swap(_M_header, __t._M_header);
        __STD::swap(_M_node_count, __t._M_node_count);
        __STD::swap(_M_key_compare, __t._M_key_compare);
      }
        
    public:
                                    // insert/erase
    	//插入节点,但是节点值必须唯一
      pair<iterator,bool> insert_unique(const value_type& __x);
      //插入节点,节点值可以与当前RB-Tree节点值相等
      iterator insert_equal(const value_type& __x);
    
      //在指定位置插入节点
      iterator insert_unique(iterator __position, const value_type& __x);
      iterator insert_equal(iterator __position, const value_type& __x);
    
    #ifdef __STL_MEMBER_TEMPLATES  
      template <class _InputIterator>
      void insert_unique(_InputIterator __first, _InputIterator __last);
      template <class _InputIterator>
      void insert_equal(_InputIterator __first, _InputIterator __last);
    #else /* __STL_MEMBER_TEMPLATES */
      void insert_unique(const_iterator __first, const_iterator __last);
      void insert_unique(const value_type* __first, const value_type* __last);
      void insert_equal(const_iterator __first, const_iterator __last);
      void insert_equal(const value_type* __first, const value_type* __last);
    #endif /* __STL_MEMBER_TEMPLATES */
    
      //删除节点
      void erase(iterator __position);
      size_type erase(const key_type& __x);
      void erase(iterator __first, iterator __last);
      void erase(const key_type* __first, const key_type* __last);
      //清除RB-Tree
      void clear() {
        if (_M_node_count != 0) {
          _M_erase(_M_root());
          _M_leftmost() = _M_header;
          _M_root() = 0;
          _M_rightmost() = _M_header;
          _M_node_count = 0;
        }
      }      
    
    public:
                                    // set operations:
      iterator find(const key_type& __x);
      const_iterator find(const key_type& __x) const;
      size_type count(const key_type& __x) const;
      iterator lower_bound(const key_type& __x);
      const_iterator lower_bound(const key_type& __x) const;
      iterator upper_bound(const key_type& __x);
      const_iterator upper_bound(const key_type& __x) const;
      pair<iterator,iterator> equal_range(const key_type& __x);
      pair<const_iterator, const_iterator> equal_range(const key_type& __x) const;
    
    public:
                                    // Debugging.
      bool __rb_verify() const;
    };
    
    //以下是操作符重载
    //重载operator==运算符，使用的是STL泛型算法
    template <class _Key, class _Value, class _KeyOfValue, 
              class _Compare, class _Alloc>
    inline bool 
    operator==(const _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>& __x, 
               const _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>& __y)
    {
      return __x.size() == __y.size() &&
    	  //STL的算法equal(__x.begin(), __x.end(), __y.begin());
             equal(__x.begin(), __x.end(), __y.begin());
    }
     //重载operator<运算符，使用的是STL泛型算法
    template <class _Key, class _Value, class _KeyOfValue, 
              class _Compare, class _Alloc>
    inline bool 
    operator<(const _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>& __x, 
              const _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>& __y)
    {
      return lexicographical_compare(__x.begin(), __x.end(), 
                                     __y.begin(), __y.end());
    }
    
    #ifdef __STL_FUNCTION_TMPL_PARTIAL_ORDER
    
    template <class _Key, class _Value, class _KeyOfValue, 
              class _Compare, class _Alloc>
    inline bool 
    operator!=(const _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>& __x, 
               const _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>& __y) {
      return !(__x == __y);
    }
    
    template <class _Key, class _Value, class _KeyOfValue, 
              class _Compare, class _Alloc>
    inline bool 
    operator>(const _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>& __x, 
              const _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>& __y) {
      return __y < __x;
    }
    
    template <class _Key, class _Value, class _KeyOfValue, 
              class _Compare, class _Alloc>
    inline bool 
    operator<=(const _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>& __x, 
               const _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>& __y) {
      return !(__y < __x);
    }
    
    template <class _Key, class _Value, class _KeyOfValue, 
              class _Compare, class _Alloc>
    inline bool 
    operator>=(const _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>& __x, 
               const _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>& __y) {
      return !(__x < __y);
    }
    
    
    template <class _Key, class _Value, class _KeyOfValue, 
              class _Compare, class _Alloc>
    inline void 
    swap(_Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>& __x, 
         _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>& __y)
    {
      __x.swap(__y);
    }
    
    #endif /* __STL_FUNCTION_TMPL_PARTIAL_ORDER */
    
    
    template <class _Key, class _Value, class _KeyOfValue, 
              class _Compare, class _Alloc>
    _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>& 
    _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>
      ::operator=(const _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>& __x)
    {
      if (this != &__x) {
                                    // Note that _Key may be a constant type.
        clear();
        _M_node_count = 0;
        _M_key_compare = __x._M_key_compare;        
        if (__x._M_root() == 0) {
          _M_root() = 0;
          _M_leftmost() = _M_header;
          _M_rightmost() = _M_header;
        }
        else {
          _M_root() = _M_copy(__x._M_root(), _M_header);
          _M_leftmost() = _S_minimum(_M_root());
          _M_rightmost() = _S_maximum(_M_root());
          _M_node_count = __x._M_node_count;
        }
      }
      return *this;
    }
    
    
    template <class _Key, class _Value, class _KeyOfValue, 
              class _Compare, class _Alloc>
    typename _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>::iterator
    _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>
      ::_M_insert(_Base_ptr __x_, _Base_ptr __y_, const _Value& __v)
    {//参数x_为新值插入点，参数y_为插入点之父节点，参数v 为新值 
      _Link_type __x = (_Link_type) __x_;
      _Link_type __y = (_Link_type) __y_;
      _Link_type __z;
    
      if (__y == _M_header || __x != 0 || 
          _M_key_compare(_KeyOfValue()(__v), _S_key(__y))) {
        __z = _M_create_node(__v);//创建值为v的节点z
        _S_left(__y) = __z;               // also makes _M_leftmost() = __z 
                                          //    when __y == _M_header
        if (__y == _M_header) {
          _M_root() = __z;
          _M_rightmost() = __z;
        }
        else if (__y == _M_leftmost())//若y为最左节点
          _M_leftmost() = __z;   // maintain _M_leftmost() pointing to min node
      }
      else {
        __z = _M_create_node(__v);
        _S_right(__y) = __z;
        if (__y == _M_rightmost())
          _M_rightmost() = __z;  // maintain _M_rightmost() pointing to max node
      }
      _S_parent(__z) = __y;//设定新节点的父节点
      _S_left(__z) = 0;//设定新节点的左孩子
      _S_right(__z) = 0;//设定新节点的右孩子
      _Rb_tree_rebalance(__z, _M_header->_M_parent);//调整RB-Tree使其满足性质
      ++_M_node_count;//节点数增加1
      return iterator(__z);//返回新节点迭代器
    }
    
    template <class _Key, class _Value, class _KeyOfValue, 
              class _Compare, class _Alloc>
    typename _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>::iterator
    _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>
      ::insert_equal(const _Value& __v)
    {
      _Link_type __y = _M_header;
      _Link_type __x = _M_root();//从根节点开始
      while (__x != 0) {//从根节点开始,往下寻找合适插入点
        __y = __x;
    	//判断新插入节点值与当前节点x值的大小,以便判断往x的左边走还是往右边走
        __x = _M_key_compare(_KeyOfValue()(__v), _S_key(__x)) ? 
                _S_left(__x) : _S_right(__x);
      }
      return _M_insert(__x, __y, __v);
    }
    
    
    // 安插新值；节点键值不允许重复，若重复则安插无效。  
    // 注意，传回值是个pair，第一元素是个 RB-tree 迭代器，指向新增节点，  
    // 第二元素表示安插成功与否。
    template <class _Key, class _Value, class _KeyOfValue, 
              class _Compare, class _Alloc>
    pair<typename _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>::iterator, 
         bool>
    _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>
      ::insert_unique(const _Value& __v)
    {
      _Link_type __y = _M_header;
      _Link_type __x = _M_root();//从根节点开始
      bool __comp = true;
      while (__x != 0) {//从根节点开始,往下寻找合适插入点
        __y = __x;
    	//判断新插入节点值与当前节点x值的大小,以便判断往x的左边走还是往右边走
        __comp = _M_key_compare(_KeyOfValue()(__v), _S_key(__x));
        __x = __comp ? _S_left(__x) : _S_right(__x);
      }
      //离开while循环之后，y所指即为安插点的父节点，x必为叶子节点
      iterator __j = iterator(__y);//令迭代器j指向插入节点之父节点y   
      if (__comp)//若为真
        if (__j == begin())//若插入点之父节点为最左节点     
          return pair<iterator,bool>(_M_insert(__x, __y, __v), true);
        else//否则(插入点之父节点不在最左节点)
          --__j;//调整j
       // 小于新值（表示遇「小」，将安插于右侧）  
      if (_M_key_compare(_S_key(__j._M_node), _KeyOfValue()(__v)))
        return pair<iterator,bool>(_M_insert(__x, __y, __v), true);
      //若运行到这里，表示键值有重复，不应该插入 
      return pair<iterator,bool>(__j, false);
    }
    
    
    template <class _Key, class _Val, class _KeyOfValue, 
              class _Compare, class _Alloc>
    typename _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>::iterator 
    _Rb_tree<_Key, _Val, _KeyOfValue, _Compare, _Alloc>
      ::insert_unique(iterator __position, const _Val& __v)
    {
      if (__position._M_node == _M_header->_M_left) { // begin()
        if (size() > 0 && 
            _M_key_compare(_KeyOfValue()(__v), _S_key(__position._M_node)))
          return _M_insert(__position._M_node, __position._M_node, __v);
        // first argument just needs to be non-null 
        else
          return insert_unique(__v).first;
      } else if (__position._M_node == _M_header) { // end()
        if (_M_key_compare(_S_key(_M_rightmost()), _KeyOfValue()(__v)))
          return _M_insert(0, _M_rightmost(), __v);
        else
          return insert_unique(__v).first;
      } else {
        iterator __before = __position;
        --__before;
        if (_M_key_compare(_S_key(__before._M_node), _KeyOfValue()(__v)) 
            && _M_key_compare(_KeyOfValue()(__v), _S_key(__position._M_node))) {
          if (_S_right(__before._M_node) == 0)
            return _M_insert(0, __before._M_node, __v); 
          else
            return _M_insert(__position._M_node, __position._M_node, __v);
        // first argument just needs to be non-null 
        } else
          return insert_unique(__v).first;
      }
    }
    
    template <class _Key, class _Val, class _KeyOfValue, 
              class _Compare, class _Alloc>
    typename _Rb_tree<_Key,_Val,_KeyOfValue,_Compare,_Alloc>::iterator 
    _Rb_tree<_Key,_Val,_KeyOfValue,_Compare,_Alloc>
      ::insert_equal(iterator __position, const _Val& __v)
    {
      if (__position._M_node == _M_header->_M_left) { // begin()
        if (size() > 0 && 
            !_M_key_compare(_S_key(__position._M_node), _KeyOfValue()(__v)))
          return _M_insert(__position._M_node, __position._M_node, __v);
        // first argument just needs to be non-null 
        else
          return insert_equal(__v);
      } else if (__position._M_node == _M_header) {// end()
        if (!_M_key_compare(_KeyOfValue()(__v), _S_key(_M_rightmost())))
          return _M_insert(0, _M_rightmost(), __v);
        else
          return insert_equal(__v);
      } else {
        iterator __before = __position;
        --__before;
        if (!_M_key_compare(_KeyOfValue()(__v), _S_key(__before._M_node))
            && !_M_key_compare(_S_key(__position._M_node), _KeyOfValue()(__v))) {
          if (_S_right(__before._M_node) == 0)
            return _M_insert(0, __before._M_node, __v); 
          else
            return _M_insert(__position._M_node, __position._M_node, __v);
        // first argument just needs to be non-null 
        } else
          return insert_equal(__v);
      }
    }
    
    #ifdef __STL_MEMBER_TEMPLATES  
    
    template <class _Key, class _Val, class _KoV, class _Cmp, class _Alloc>
      template<class _II>
    void _Rb_tree<_Key,_Val,_KoV,_Cmp,_Alloc>
      ::insert_equal(_II __first, _II __last)
    {
      for ( ; __first != __last; ++__first)
        insert_equal(*__first);
    }
    
    template <class _Key, class _Val, class _KoV, class _Cmp, class _Alloc> 
      template<class _II>
    void _Rb_tree<_Key,_Val,_KoV,_Cmp,_Alloc>
      ::insert_unique(_II __first, _II __last) {
      for ( ; __first != __last; ++__first)
        insert_unique(*__first);
    }
    
    #else /* __STL_MEMBER_TEMPLATES */
    
    template <class _Key, class _Val, class _KoV, class _Cmp, class _Alloc>
    void
    _Rb_tree<_Key,_Val,_KoV,_Cmp,_Alloc>
      ::insert_equal(const _Val* __first, const _Val* __last)
    {
      for ( ; __first != __last; ++__first)
        insert_equal(*__first);
    }
    
    template <class _Key, class _Val, class _KoV, class _Cmp, class _Alloc>
    void
    _Rb_tree<_Key,_Val,_KoV,_Cmp,_Alloc>
      ::insert_equal(const_iterator __first, const_iterator __last)
    {
      for ( ; __first != __last; ++__first)
        insert_equal(*__first);
    }
    
    template <class _Key, class _Val, class _KoV, class _Cmp, class _Alloc>
    void 
    _Rb_tree<_Key,_Val,_KoV,_Cmp,_Alloc>
      ::insert_unique(const _Val* __first, const _Val* __last)
    {
      for ( ; __first != __last; ++__first)
        insert_unique(*__first);
    }
    
    template <class _Key, class _Val, class _KoV, class _Cmp, class _Alloc>
    void _Rb_tree<_Key,_Val,_KoV,_Cmp,_Alloc>
      ::insert_unique(const_iterator __first, const_iterator __last)
    {
      for ( ; __first != __last; ++__first)
        insert_unique(*__first);
    }
    
    #endif /* __STL_MEMBER_TEMPLATES */
             
    template <class _Key, class _Value, class _KeyOfValue, 
              class _Compare, class _Alloc>
    inline void _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>
      ::erase(iterator __position)
    {
      _Link_type __y = 
        (_Link_type) _Rb_tree_rebalance_for_erase(__position._M_node,
                                                  _M_header->_M_parent,
                                                  _M_header->_M_left,
                                                  _M_header->_M_right);
      destroy_node(__y);
      --_M_node_count;
    }
    
    template <class _Key, class _Value, class _KeyOfValue, 
              class _Compare, class _Alloc>
    typename _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>::size_type 
    _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>::erase(const _Key& __x)
    {
      pair<iterator,iterator> __p = equal_range(__x);
      size_type __n = 0;
      distance(__p.first, __p.second, __n);
      erase(__p.first, __p.second);
      return __n;
    }
    
    template <class _Key, class _Val, class _KoV, class _Compare, class _Alloc>
    typename _Rb_tree<_Key, _Val, _KoV, _Compare, _Alloc>::_Link_type 
    _Rb_tree<_Key,_Val,_KoV,_Compare,_Alloc>
      ::_M_copy(_Link_type __x, _Link_type __p)
    {
                            // structural copy.  __x and __p must be non-null.
      _Link_type __top = _M_clone_node(__x);
      __top->_M_parent = __p;
     
      __STL_TRY {
        if (__x->_M_right)
          __top->_M_right = _M_copy(_S_right(__x), __top);
        __p = __top;
        __x = _S_left(__x);
    
        while (__x != 0) {
          _Link_type __y = _M_clone_node(__x);
          __p->_M_left = __y;
          __y->_M_parent = __p;
          if (__x->_M_right)
            __y->_M_right = _M_copy(_S_right(__x), __y);
          __p = __y;
          __x = _S_left(__x);
        }
      }
      __STL_UNWIND(_M_erase(__top));
    
      return __top;
    }
    
    template <class _Key, class _Value, class _KeyOfValue, 
              class _Compare, class _Alloc>
    void _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>
      ::_M_erase(_Link_type __x)
    {
                                    // erase without rebalancing
      while (__x != 0) {
        _M_erase(_S_right(__x));
        _Link_type __y = _S_left(__x);
        destroy_node(__x);
        __x = __y;
      }
    }
    
    template <class _Key, class _Value, class _KeyOfValue, 
              class _Compare, class _Alloc>
    void _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>
      ::erase(iterator __first, iterator __last)
    {
      if (__first == begin() && __last == end())
        clear();
      else
        while (__first != __last) erase(__first++);
    }
    
    template <class _Key, class _Value, class _KeyOfValue, 
              class _Compare, class _Alloc>
    void _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>
      ::erase(const _Key* __first, const _Key* __last) 
    {
      while (__first != __last) erase(*__first++);
    }
    
    template <class _Key, class _Value, class _KeyOfValue, 
              class _Compare, class _Alloc>
    typename _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>::iterator 
    _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>::find(const _Key& __k)
    {
      _Link_type __y = _M_header;      // Last node which is not less than __k. 
      _Link_type __x = _M_root();      // Current node. 
    
      while (__x != 0) 
        if (!_M_key_compare(_S_key(__x), __k))
          __y = __x, __x = _S_left(__x);
        else
          __x = _S_right(__x);
    
      iterator __j = iterator(__y);   
      return (__j == end() || _M_key_compare(__k, _S_key(__j._M_node))) ? 
         end() : __j;
    }
    
    //查找RB树中是否有键值为k的节点
    template <class _Key, class _Value, class _KeyOfValue, 
              class _Compare, class _Alloc>
    typename _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>::const_iterator 
    _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>::find(const _Key& __k) const
    {
      _Link_type __y = _M_header; /* Last node which is not less than __k. */
      _Link_type __x = _M_root(); /* Current node. */
    
      while (__x != 0) {
        if (!_M_key_compare(_S_key(__x), __k))//若k比当前节点x键值小
          __y = __x, __x = _S_left(__x);
        else
          __x = _S_right(__x);
      }
      const_iterator __j = const_iterator(__y);   
      return (__j == end() || _M_key_compare(__k, _S_key(__j._M_node))) ?
        end() : __j;
    }
    
    //计算RB树中键值为k的节点的个数 
    template <class _Key, class _Value, class _KeyOfValue, 
              class _Compare, class _Alloc>
    typename _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>::size_type 
    _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>
      ::count(const _Key& __k) const
    {
      pair<const_iterator, const_iterator> __p = equal_range(__k);
      size_type __n = 0;
      distance(__p.first, __p.second, __n);
      return __n;
    }
    
    template <class _Key, class _Value, class _KeyOfValue, 
              class _Compare, class _Alloc>
    typename _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>::iterator 
    _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>
      ::lower_bound(const _Key& __k)
    {
      _Link_type __y = _M_header; /* Last node which is not less than __k. */
      _Link_type __x = _M_root(); /* Current node. */
    
      while (__x != 0) 
        if (!_M_key_compare(_S_key(__x), __k))
          __y = __x, __x = _S_left(__x);
        else
          __x = _S_right(__x);
    
      return iterator(__y);
    }
    
    template <class _Key, class _Value, class _KeyOfValue, 
              class _Compare, class _Alloc>
    typename _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>::const_iterator 
    _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>
      ::lower_bound(const _Key& __k) const
    {
      _Link_type __y = _M_header; /* Last node which is not less than __k. */
      _Link_type __x = _M_root(); /* Current node. */
    
      while (__x != 0) 
        if (!_M_key_compare(_S_key(__x), __k))
          __y = __x, __x = _S_left(__x);
        else
          __x = _S_right(__x);
    
      return const_iterator(__y);
    }
    
    template <class _Key, class _Value, class _KeyOfValue, 
              class _Compare, class _Alloc>
    typename _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>::iterator 
    _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>
      ::upper_bound(const _Key& __k)
    {
      _Link_type __y = _M_header; /* Last node which is greater than __k. */
      _Link_type __x = _M_root(); /* Current node. */
    
       while (__x != 0) 
         if (_M_key_compare(__k, _S_key(__x)))
           __y = __x, __x = _S_left(__x);
         else
           __x = _S_right(__x);
    
       return iterator(__y);
    }
    
    template <class _Key, class _Value, class _KeyOfValue, 
              class _Compare, class _Alloc>
    typename _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>::const_iterator 
    _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>
      ::upper_bound(const _Key& __k) const
    {
      _Link_type __y = _M_header; /* Last node which is greater than __k. */
      _Link_type __x = _M_root(); /* Current node. */
    
       while (__x != 0) 
         if (_M_key_compare(__k, _S_key(__x)))
           __y = __x, __x = _S_left(__x);
         else
           __x = _S_right(__x);
    
       return const_iterator(__y);
    }
    
    template <class _Key, class _Value, class _KeyOfValue, 
              class _Compare, class _Alloc>
    inline 
    pair<typename _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>::iterator,
         typename _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>::iterator>
    _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>
      ::equal_range(const _Key& __k)
    {
      return pair<iterator, iterator>(lower_bound(__k), upper_bound(__k));
    }
    
    template <class _Key, class _Value, class _KoV, class _Compare, class _Alloc>
    inline 
    pair<typename _Rb_tree<_Key, _Value, _KoV, _Compare, _Alloc>::const_iterator,
         typename _Rb_tree<_Key, _Value, _KoV, _Compare, _Alloc>::const_iterator>
    _Rb_tree<_Key, _Value, _KoV, _Compare, _Alloc>
      ::equal_range(const _Key& __k) const
    {
      return pair<const_iterator,const_iterator>(lower_bound(__k),
                                                 upper_bound(__k));
    }
    
    //计算从 node 至 root路径中的黑节点数量 
    inline int 
    __black_count(_Rb_tree_node_base* __node, _Rb_tree_node_base* __root)
    {
      if (__node == 0)
        return 0;
      else {
        int __bc = __node->_M_color == _S_rb_tree_black ? 1 : 0;//若节点node为黑色,则bc为1
        if (__node == __root)//判断node是否为根节点
          return __bc;
        else
          return __bc + __black_count(__node->_M_parent, __root);//递归调用
      }
    }
    
    //验证己生这棵树是否符合RB树条件
    template <class _Key, class _Value, class _KeyOfValue, 
              class _Compare, class _Alloc>
    bool _Rb_tree<_Key,_Value,_KeyOfValue,_Compare,_Alloc>::__rb_verify() const
    {
    	//空树
      if (_M_node_count == 0 || begin() == end())
        return _M_node_count == 0 && begin() == end() &&
          _M_header->_M_left == _M_header && _M_header->_M_right == _M_header;
      
      //最左节点到根节点的黑色节点数
      int __len = __black_count(_M_leftmost(), _M_root());
      //一下走访整个RB树，针对每个节点（从最小到最大）……
      for (const_iterator __it = begin(); __it != end(); ++__it) {
        _Link_type __x = (_Link_type) __it._M_node;
        _Link_type __L = _S_left(__x);
        _Link_type __R = _S_right(__x);
    
        if (__x->_M_color == _S_rb_tree_red)//违背性质4
    		//如果一个节点是红色的，则它的两个孩子节点都是黑色的。
          if ((__L && __L->_M_color == _S_rb_tree_red) ||
              (__R && __R->_M_color == _S_rb_tree_red))
            return false;
    
        //以下是违背二叉查找树性质
    	//节点的左孩子节点键值小于该节点键值
    	//节点的右孩子节点键值大于该节点键值
    	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;
    
    	//[叶子结点到root]路径内的黑色节点数，与[最左节点至root]路径内的黑色节点不同。不符合RB树要求 
    	//违背性质5
        if (!__L && !__R && __black_count(__x, _M_root()) != __len)
          return false;
      }
    
      if (_M_leftmost() != _Rb_tree_node_base::_S_minimum(_M_root()))
        return false; // 最左节点不为最小节点，不符合二叉查找树的要求
      if (_M_rightmost() != _Rb_tree_node_base::_S_maximum(_M_root()))
        return false;// 最右节点不为最大节点，不符不符合二叉查找树的要求
    
      return true;
    }
    
    // Class rb_tree is not part of the C++ standard.  It is provided for
    // compatibility with the HP STL.
    
    template <class _Key, class _Value, class _KeyOfValue, class _Compare,
              class _Alloc = __STL_DEFAULT_ALLOCATOR(_Value) >
    struct rb_tree : public _Rb_tree<_Key, _Value, _KeyOfValue, _Compare, _Alloc>
    {
      typedef _Rb_tree<_Key, _Value, _KeyOfValue, _Compare, _Alloc> _Base;
      typedef typename _Base::allocator_type allocator_type;
    
      rb_tree(const _Compare& __comp = _Compare(),
              const allocator_type& __a = allocator_type())
        : _Base(__comp, __a) {}
      
      ~rb_tree() {}
    };
    
    #if defined(__sgi) && !defined(__GNUC__) && (_MIPS_SIM != _MIPS_SIM_ABI32)
    #pragma reset woff 1375
    #endif
    
    __STL_END_NAMESPACE 
    
    #endif /* __SGI_STL_INTERNAL_TREE_H */
    
    // Local Variables:
    // mode:C++
    // End:

参考资料：

《STL源码剖析》侯捷

《[STL源码剖析---stl\_tree.h阅读笔记][STL_---stl_tree.h]》

[RB-Tree]: http://blog.csdn.net/chenhanzhun/article/details/38405041
[Link 1]: http://blog.csdn.net/chenhanzhun/article/details/21990545
[SouthEast]: /images/20220810/87c337887517411bb5212d225a690bc0.png
[STL_---stl_tree.h]: http://blog.csdn.net/kangroger/article/details/38587785