STL源码：红黑树

悠悠 2022-08-07 09:52 138阅读 0赞

# 红黑树的性质和插入操作 #

这部分参考文章 《[红黑树操作及实现》][Link 1]

# 红黑树节点结构 #

typedef bool __rb_tree_color_type;
    const __rb_tree_color_type __rb_tree_red = false;     // 红色为0
    const __rb_tree_color_type __rb_tree_black = true; // 黑色为1
    
    struct __rb_tree_node_base
    {
      typedef __rb_tree_color_type color_type;
      typedef __rb_tree_node_base* base_ptr;
    
      color_type color;     // 节点颜色，红色或黑色
      base_ptr parent;      // 该指针指向其父节点
      base_ptr left;        // 指向左节点
      base_ptr right;       // 指向右节点
    
      //二叉树搜索树，一直向左走，找到最小的值
      static base_ptr minimum(base_ptr x)
      {
    	 while (x->left != 0) x = x->left; 
    	 return x;                            
      }
    
      //二叉搜索树，一直向右走，找最大的值
      static base_ptr maximum(base_ptr x)
      {
        while (x->right != 0) x = x->right; 
        return x;                           
      }
    };
    
    template <class Value>
    struct __rb_tree_node : public __rb_tree_node_base
    {
      typedef __rb_tree_node<Value>* link_type;
      Value value_field;   //节点的值
    };

红黑树节点的结构如下：

![Center][]

# 红黑树的迭代器 #

RB-tree迭代器实现分为两层，这种设计理念类似于slist。下图是两层节点结构和双层迭代器结构间的关系，其中主要意义是：\_\_rb\_tree\_node继承自\_\_rb\_tree\_node\_base，\_\_rb\_tree\_iterator继承自\_\_rb\_tree\_base\_iterator。

**这里主要关注底层迭代器increment()和decrement()这两个操作分别实现迭代器的自加和自减操作，其前进后退举动完全依据二叉搜索树寻找前驱后继节点的规则（二叉搜索树寻找前驱后继规则[参考该文章][Link 2]），只是多了一些实现技巧**。

代码的实现见最后的完整源代码。这里的实现技巧如下。每当插入新节点时，不但要依据红黑树规则来调整，还有维护header的正确性，使其父节点指向根节点，左子节点指向最小节点，右子节点指向最大节点。

![20150330201842035][]

# 红黑树插入操作 #

提供了两种插入操作：insert\_unique()和insert\_equal()，分别在树中插入独一无二的值、插入可以重复的值。有多个版本，重点分析insert\_unique(const Value& v)版本。

真正指向插入操作的是\_\_insert()函数。

# 红黑树源码 #

//stl_tree.h
    #ifndef __SGI_STL_INTERNAL_TREE_H
    #define __SGI_STL_INTERNAL_TREE_H
    
    
    /*
    Red-black tree（红黑树）class，用来当做SLT关联容器的底层机制（如set，multiset，map，
    multimap）。里面所用的insertion和deletion方法以Cormen, Leiserson 和 Riveset所著的
    《算法导论》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）可以有线性时间实现。
    (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.
    */
    #include <stl_algobase.h>
    #include <stl_alloc.h>
    #include <stl_construct.h>
    #include <stl_function.h>
    
    __STL_BEGIN_NAMESPACE 
    //定义红色黑色。红色为0，黑色为1
    typedef bool __rb_tree_color_type;
    const __rb_tree_color_type __rb_tree_red = false;
    const __rb_tree_color_type __rb_tree_black = true;
    //红黑树Base类
    struct __rb_tree_node_base
    {
      typedef __rb_tree_color_type color_type;
      typedef __rb_tree_node_base* base_ptr;
    
      color_type color; 	// 节点颜色，红或者黑
      base_ptr parent;  	// RB树的许多操作，必须知道其父结点
      base_ptr left;	  	// 指向左孩子节点。
      base_ptr right;   	// 指向右孩子节点。
    
      static base_ptr minimum(base_ptr x)
      {
        while (x->left != 0) x = x->left;	// 一直向左走，就会找到最小值
        return x;							// 这是二叉查找树的性质。同理下面的函数
      }
    
      static base_ptr maximum(base_ptr x)
      {
        while (x->right != 0) x = x->right;
        return x;
      }
    };
    
    //红黑树类，继承Base类
    template <class Value>
    struct __rb_tree_node : public __rb_tree_node_base
    {
      typedef __rb_tree_node<Value>* link_type;//指向节点的指针
      Value value_field;	// 节点的值
    };
    //迭代器基类,类型为bidirectional_iterator_tag，可以双向移动
    struct __rb_tree_base_iterator
    {
      typedef __rb_tree_node_base::base_ptr base_ptr;//指向红黑树节点指针
      typedef bidirectional_iterator_tag iterator_category;
      typedef ptrdiff_t difference_type;
    
      //指向红黑树节点的指针，用它来和容器产生关系
      base_ptr node;
    
      /*
    	重载运算符++和--。目的是找到前驱和后继节点。
    	关于前驱和后继节点的定义，类似二叉查找树。可以在这里找到：
    	http://blog.csdn.net/u013074465/article/details/41699891
      */
      //下面只是为了实现oprerator++的，其他地方不会调用了。
      //++是找到其后继节点
      void increment()
      {
    	//如果有右孩子，就是找右子树的最小值
        if (node->right != 0) {		// 如果有右孩子
          node = node->right;		// 就向右走
          while (node->left != 0)	// 然后向左走到底
            node = node->left;		
        }
    	//如果无右子树。那么就找其最低祖先节点，且这个最低祖先节点的左孩子节点
    	//也是其祖先节点（每个节点就是自己的祖先节点）
        else {					// 没有右孩子
          base_ptr y = node->parent;	// 找出父节点
          while (node == y->right) {	// 如果现行节点本身是个右子节点
            node = y;				// 就一直上溯，直到「不为右子节点」止。
            y = y->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.
    		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，在上面有图
    	  */
          if (node->right != y)		// 若此时的右子节点不等于此时的父节点
            node = y;				// 此时的父节点即为解答
                                    // 否则此时的node为解答
        }						
      
      }
    
       //查找前驱结点。
      void decrement()
      {
        if (node->color == __rb_tree_red &&	// 如果是红节点，且
            node->parent->parent == node)		// 父节点的父节点等于自己
          node = node->right;				// 状况(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。上面有图
    	  */
    	//左子树的最大值结点
        else if (node->left != 0) {	
          base_ptr y = node->left;
          while (y->right != 0)	
            y = y->right;	
          node = y;		
        }
    	/*
    	既非根节点，且无左子树。找其最低祖先节点y，且y的右孩子也是其祖先节点
    	*/
        else {							
          base_ptr y = node->parent;			//找出父节点
          while (node == y->left) {	
            node = y;					
            y = y->parent;	
          }
          node = y;
        }
      }
    };
    //此处为迭代器
    template <class Value, class Ref, class Ptr>
    struct __rb_tree_iterator : public __rb_tree_base_iterator
    {
      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;
    
      //几个构造函数
      __rb_tree_iterator() {}
      __rb_tree_iterator(link_type x) { node = x; }
      __rb_tree_iterator(const iterator& it) { node = it.node; }
    
      //重载操作符
      reference operator*() const { return link_type(node)->value_field; }
    #ifndef __SGI_STL_NO_ARROW_OPERATOR
      pointer operator->() const { return &(operator*()); }
    #endif /* __SGI_STL_NO_ARROW_OPERATOR */
    
    	//++做了封装，调用的是increment()
      self& operator++() { increment(); return *this; }
      self operator++(int) {
        self tmp = *this;
        increment();
        return tmp;
      }
        //调用的是decrement
      self& operator--() { decrement(); return *this; }
      self operator--(int) {
        self tmp = *this;
        decrement();
        return tmp;
      }
    };
    //两个迭代器相等，意味着它们指向同一个红黑树节点
    inline bool operator==(const __rb_tree_base_iterator& x,
                           const __rb_tree_base_iterator& y) {
      return x.node == y.node;
    }
    
    inline bool operator!=(const __rb_tree_base_iterator& x,
                           const __rb_tree_base_iterator& y) {
      return x.node != y.node;
    }
    
    #ifndef __STL_CLASS_PARTIAL_SPECIALIZATION
    //返回迭代器类型
    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（3）。此时要旋转和改变颜色
    */
    //左旋转
    inline void 
    __rb_tree_rotate_left(__rb_tree_node_base* x, __rb_tree_node_base*& root)
    {
      // x 为旋转点
      __rb_tree_node_base* y = x->right;	// y为x的右孩子
      x->right = y->left;
      if (y->left !=0)
        y->left->parent = x;		// 不要忘了回马枪设置父节点
      y->parent = x->parent;
    
      // 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)					// x 为根节点
        root = y;
      else if (x == x->parent->left)	// x 为父节点的左孩子
        x->parent->left = y;
      else							// x 为父节点的右孩子
        x->parent->right = y;			
      y->left = x;
      x->parent = y;
    }
    
    //右旋转
    inline void 
    __rb_tree_rotate_right(__rb_tree_node_base* x, __rb_tree_node_base*& root)
    {
      // x 为旋转点
      __rb_tree_node_base* y = x->left;	// y x的左孩子
      x->left = y->right;
      if (y->right != 0)
        y->right->parent = x; 	// 別忘了回马枪设置父节点
      y->parent = x->parent;
    
      // 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)					// x 为根节点
        root = y;
      else if (x == x->parent->right)	// x 为父节点的右孩子
        x->parent->right = y;
      else							// x 为父节点的左孩子
        x->parent->left = y;
      y->right = x;
      x->parent = y;
    }
    
    
    //重新令RB-tree平衡（改变颜色和旋转）参数x为新增节点，参数二为root节点
    inline void 
    __rb_tree_rebalance(__rb_tree_node_base* x, __rb_tree_node_base*& root)
    {
      x->color = __rb_tree_red;		// 新节点比为红色
      while (x != root && x->parent->color == __rb_tree_red) { // 父节点为红色
        if (x->parent == x->parent->parent->left) { // 父节点为祖父节点的左孩子
          __rb_tree_node_base* y = x->parent->parent->right;	// 令y 为伯父节点
          if (y && y->color == __rb_tree_red) { 		// 伯父节点存在，且为红色
            x->parent->color = __rb_tree_black;  		// 更改父节点为黑色
            y->color = __rb_tree_black;				// 更改伯父节点为黑色
            x->parent->parent->color = __rb_tree_red; 	// 更改祖父节点为红色
            x = x->parent->parent;
          }
          else {	// 无伯父节点或伯父节点为黑色（NULL就是黑色）
            if (x == x->parent->right) { // 新增节点为父节点的右孩子
              x = x->parent;
              __rb_tree_rotate_left(x, root); // 第一个参数为左旋转点
            }
            x->parent->color = __rb_tree_black;	// 改变颜色，父节点为黑色
            x->parent->parent->color = __rb_tree_red;
            __rb_tree_rotate_right(x->parent->parent, root); // 第一参数为右旋转点
          }
        }
        else {	// 父节点为祖父节点的右孩子
          __rb_tree_node_base* y = x->parent->parent->left; // y为伯父节点
          if (y && y->color == __rb_tree_red) {		// 有伯父节点且为红色
            x->parent->color = __rb_tree_black;		// 更改父节点为黑色
            y->color = __rb_tree_black; 				// 更改伯父节点为黑色
            x->parent->parent->color = __rb_tree_red; 	// 更改祖父节点为红色
            x = x->parent->parent;	// 准备继续往上层检查……
          }
          else {	// 无伯父节点或伯父节点为黑色（NULL就是黑色）
            if (x == x->parent->left) {	// 新节点为父节点的左孩子
              x = x->parent;
              __rb_tree_rotate_right(x, root); 	// 第一个参数右旋转
            }
            x->parent->color = __rb_tree_black;	// 改变颜色，父节点为黑色
            x->parent->parent->color = __rb_tree_red;
            __rb_tree_rotate_left(x->parent->parent, root); // 第一个参数做旋转
          }
        }
      }	// while 結束
      root->color = __rb_tree_black;	// 根节点永远为黑色
    }
    //删除结点z
    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->left == 0)             // z has at most one non-null child. y == z.
        x = y->right;               // x might be null.
      else
        if (y->right == 0)          // z has exactly one non-null child.  y == z.
          x = y->left;              // x is not null.
        else {                      // z has two non-null children.  Set y to
          y = y->right;             //   z's successor.  x might be null.
          while (y->left != 0)
            y = y->left;
          x = y->right;
        }
      if (y != z) {                 // relink y in place of z.  y is z's successor
        z->left->parent = y; 
        y->left = z->left;
        if (y != z->right) {
          x_parent = y->parent;
          if (x) x->parent = y->parent;
          y->parent->left = x;      // y must be a left child
          y->right = z->right;
          z->right->parent = y;
        }
        else
          x_parent = y;  
        if (root == z)
          root = y;
        else if (z->parent->left == z)
          z->parent->left = y;
        else 
          z->parent->right = y;
        y->parent = z->parent;
        __STD::swap(y->color, z->color);
        y = z;
        // y now points to node to be actually deleted
      }
      else {                        // y == z
        x_parent = y->parent;
        if (x) x->parent = y->parent;   
        if (root == z)
          root = x;
        else 
          if (z->parent->left == z)
            z->parent->left = x;
          else
            z->parent->right = x;
        if (leftmost == z) 
          if (z->right == 0)        // z->left must be null also
            leftmost = z->parent;
        // makes leftmost == header if z == root
          else
            leftmost = __rb_tree_node_base::minimum(x);
        if (rightmost == z)  
          if (z->left == 0)         // z->right must be null also
            rightmost = z->parent;  
        // makes rightmost == header if z == root
          else                      // x == z->left
            rightmost = __rb_tree_node_base::maximum(x);
      }
      if (y->color != __rb_tree_red) { 
        while (x != root && (x == 0 || x->color == __rb_tree_black))
          if (x == x_parent->left) {
            __rb_tree_node_base* w = x_parent->right;
            if (w->color == __rb_tree_red) {
              w->color = __rb_tree_black;
              x_parent->color = __rb_tree_red;
              __rb_tree_rotate_left(x_parent, root);
              w = x_parent->right;
            }
            if ((w->left == 0 || w->left->color == __rb_tree_black) &&
                (w->right == 0 || w->right->color == __rb_tree_black)) {
              w->color = __rb_tree_red;
              x = x_parent;
              x_parent = x_parent->parent;
            } else {
              if (w->right == 0 || w->right->color == __rb_tree_black) {
                if (w->left) w->left->color = __rb_tree_black;
                w->color = __rb_tree_red;
                __rb_tree_rotate_right(w, root);
                w = x_parent->right;
              }
              w->color = x_parent->color;
              x_parent->color = __rb_tree_black;
              if (w->right) w->right->color = __rb_tree_black;
              __rb_tree_rotate_left(x_parent, root);
              break;
            }
          } else {                  // same as above, with right <-> left.
            __rb_tree_node_base* w = x_parent->left;
            if (w->color == __rb_tree_red) {
              w->color = __rb_tree_black;
              x_parent->color = __rb_tree_red;
              __rb_tree_rotate_right(x_parent, root);
              w = x_parent->left;
            }
            if ((w->right == 0 || w->right->color == __rb_tree_black) &&
                (w->left == 0 || w->left->color == __rb_tree_black)) {
              w->color = __rb_tree_red;
              x = x_parent;
              x_parent = x_parent->parent;
            } else {
              if (w->left == 0 || w->left->color == __rb_tree_black) {
                if (w->right) w->right->color = __rb_tree_black;
                w->color = __rb_tree_red;
                __rb_tree_rotate_left(w, root);
                w = x_parent->left;
              }
              w->color = x_parent->color;
              x_parent->color = __rb_tree_black;
              if (w->left) w->left->color = __rb_tree_black;
              __rb_tree_rotate_right(x_parent, root);
              break;
            }
          }
        if (x) x->color = __rb_tree_black;
      }
      return y;
    }
    
    template <class Key, class Value, class KeyOfValue, class Compare,
              class Alloc = alloc>
    class rb_tree {
    protected:
      typedef void* void_pointer;
      typedef __rb_tree_node_base* base_ptr;
      typedef __rb_tree_node<Value> rb_tree_node;
      typedef simple_alloc<rb_tree_node, Alloc> rb_tree_node_allocator;
      typedef __rb_tree_color_type color_type;
    public:
      //这里没有定义iterator，在后面定义
      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;
    protected:
      link_type get_node() { return rb_tree_node_allocator::allocate(); }
      void put_node(link_type p) { rb_tree_node_allocator::deallocate(p); }
    
      link_type create_node(const value_type& x) {
        link_type tmp = get_node();			// 配置空间
        __STL_TRY {
          construct(&tmp->value_field, x);	// 构建内容
        }
        __STL_UNWIND(put_node(tmp));
        return tmp;
      }
    
      link_type clone_node(link_type x) {	// 复制一个节点（值和颜色）
        link_type tmp = create_node(x->value_field);
        tmp->color = x->color;
        tmp->left = 0;
        tmp->right = 0;
        return tmp;
      }
    
      void destroy_node(link_type p) {
        destroy(&p->value_field);		// 析构
        put_node(p);					// 释放空间
      }
    
    protected:
      // RB-tree 只以三个资料表现
      size_type node_count; // 追踪记录树的大小（节点总数）
      link_type header;  
      Compare key_compare;	 // 节点的键值比较判断准则。是个函数 function object。
    
      //以下三个函数用来方便取得header的成员
      link_type& root() const { return (link_type&) header->parent; }
      link_type& leftmost() const { return (link_type&) header->left; }
      link_type& rightmost() const { return (link_type&) header->right; }
    
      //以下六个函数用来方便取得节点x的成员。x为函数参数
      static link_type& left(link_type x) { return (link_type&)(x->left); }
      static link_type& right(link_type x) { return (link_type&)(x->right); }
      static link_type& parent(link_type x) { return (link_type&)(x->parent); }
      static reference value(link_type x) { return x->value_field; }
      static const Key& key(link_type x) { return KeyOfValue()(value(x)); }
      static color_type& color(link_type x) { return (color_type&)(x->color); }
    
      //和上面六个作用相同，注意x参数类型不同。一个是基类指针，一个是派生类指针
      static link_type& left(base_ptr x) { return (link_type&)(x->left); }
      static link_type& right(base_ptr x) { return (link_type&)(x->right); }
      static link_type& parent(base_ptr x) { return (link_type&)(x->parent); }
      static reference value(base_ptr x) { return ((link_type)x)->value_field; }
      static const Key& key(base_ptr x) { return KeyOfValue()(value(link_type(x)));} 
      static color_type& color(base_ptr x) { return (color_type&)(link_type(x)->color); }
    
      //找最大值和最小值。node class 有这个功能函数
      static link_type minimum(link_type x) { 
        return (link_type)  __rb_tree_node_base::minimum(x);
      }
      static link_type maximum(link_type x) {
        return (link_type) __rb_tree_node_base::maximum(x);
      }
    
    public:
      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:
      iterator __insert(base_ptr x, base_ptr y, const value_type& v);
      link_type __copy(link_type x, link_type p);
      void __erase(link_type x);
      void init() {
        header = get_node();	// 产生一个节点空间，令header指向它
        color(header) = __rb_tree_red; // 令 header 尾红色，用來区 header  
                                       // 和 root（在 iterator.operator++ 中）
        root() = 0;
        leftmost() = header;	// 令 header 的左孩子为自己。
        rightmost() = header;	// 令 header 的右孩子为自己。
      }
    public:
         //默认构造函数                           // allocation/deallocation
      rb_tree(const Compare& comp = Compare())
        : node_count(0), key_compare(comp) { init(); }
    
      // 以另一个 rb_tree  x 初始化
      rb_tree(const rb_tree<Key, Value, KeyOfValue, Compare, Alloc>& x) 
        : node_count(0), key_compare(x.key_compare)
      { 
        header = get_node();	
        color(header) = __rb_tree_red;	
        if (x.root() == 0) {	//  如果 x 空树
          root() = 0;
          leftmost() = header; 
          rightmost() = header; 
        }
        else {	//  x 不是空树
          __STL_TRY {
            root() = __copy(x.root(), header);		// 拷贝红黑树x 
          }
          __STL_UNWIND(put_node(header));
          leftmost() = minimum(root());	// 令 header 的左孩子为最小节点
          rightmost() = maximum(root());	// 令 header 的右孩子为最大节点
        }
        node_count = x.node_count;
      }
      ~rb_tree() {
        clear();
        put_node(header);
      }
      rb_tree<Key, Value, KeyOfValue, Compare, Alloc>& 
      operator=(const rb_tree<Key, Value, KeyOfValue, Compare, Alloc>& x);
    
    public:    
                                    // accessors:
      Compare key_comp() const { return key_compare; }
      iterator begin() { return leftmost(); }		// RB 树的起始为最左（最小节点）
      const_iterator begin() const { return leftmost(); }
      iterator end() { return header; }	// RB 树的终节点为header所指处
      const_iterator end() const { return 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());
      } 
      bool empty() const { return node_count == 0; }
      size_type size() const { return node_count; }
      size_type max_size() const { return size_type(-1); }
    
      void swap(rb_tree<Key, Value, KeyOfValue, Compare, Alloc>& t) {
    
    	//RB-tree只有三个资料表现成员，所以两颗RB-tree互换时，只需互换3个成员
        __STD::swap(header, t.header);
        __STD::swap(node_count, t.node_count);
        __STD::swap(key_compare, t.key_compare);
      }
        
    public:
                                    // insert/erase
      // 将 x 安插到 RB-tree 中（保持节点值独一无二）。
      pair<iterator,bool> insert_unique(const value_type& x);
      // 将 x 安插到 RB-tree 中（允许重复节点）
      iterator insert_equal(const value_type& x);
    
      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 */
    
      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);
      void clear() {
        if (node_count != 0) {
          __erase(root());
          leftmost() = header;
          root() = 0;
          rightmost() = header;
          node_count = 0;
        }
      }      
    
    public:
                                    // 集合（set）的各种操作行为
      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;
    };
    
    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() && equal(x.begin(), x.end(), y.begin());
    }
    //重载<运算符，使用的是STL泛型算法<span style="font-family: Arial, Helvetica, sans-serif;">lexicographical_compare</span>
    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 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 */
    
    //重载赋值运算符=
    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) {//防止自身赋值
                                    // Note that Key may be a constant type.
        clear();//先清除
        node_count = 0;
        key_compare = x.key_compare;        
        if (x.root() == 0) {
          root() = 0;
          leftmost() = header;
          rightmost() = header;
        }
        else {
          root() = __copy(x.root(), header);
          leftmost() = minimum(root());
          rightmost() = maximum(root());
          node_count = x.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>::
    __insert(base_ptr x_, base_ptr y_, const Value& v) {
    //参数x_为新值安插点，参数y_为安插点之父节点，参数v 为新值
      link_type x = (link_type) x_;
      link_type y = (link_type) y_;
      link_type z;
    
      //key_compare是键值得比较准则，是个函数或函数指针
      if (y == header || x != 0 || key_compare(KeyOfValue()(v), key(y))) {
        z = create_node(v);  // 产生一个新节点
        left(y) = z;          // 这使得当y为header时，leftmost()=z
        if (y == header) {
          root() = z;
          rightmost() = z;
        }
        else if (y == leftmost())	// 如果y为最左节点
          leftmost() = z;           	// 维护leftmost()，使它永远指向最左节点
      }
      else {
        z = create_node(v);
        right(y) = z;				// 令新节点成为安插点之父节点y的右孩子
        if (y == rightmost())
          rightmost() = z;          	// 维护rightmost()，使它永远指向最右节点
      }
      parent(z) = y;		// 设定新节点的父节点
      left(z) = 0;		// 设定新孩子节点的左孩子
      right(z) = 0; 		// 设定新孩子节点的右孩子
                              // 新节点的颜色将在 __rb_tree_rebalance() 设定并调整
      __rb_tree_rebalance(z, header->parent);	// 参数一为新增节点，参数二为root
      ++node_count;		// 节点数增加
      return iterator(z);	// 返回迭代器，指向新增节点
    }
    
    // 安插新值；允许键值重复。返回新插入节点的迭代器
    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 = header;
      link_type x = root();	 
      while (x != 0) {		// 从根节点开始，向下寻找适当安插位置
        y = x;
        x = key_compare(KeyOfValue()(v), key(x)) ? left(x) : right(x);
      }
      return __insert(x, y, v);
    }
    
    /*
    不允许键值重复，否则安插无效。
    返回值是个pair，第一个元素是个RB-tree迭代器，指向新增节点。
    第二个元素表示安插是否成功。
    */
    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 = header;
      link_type x = root();  //从根节点开始
      bool comp = true;
      while (x != 0) { 		// 从根节点开始向下寻找适当安插位置
        y = x;
        comp = key_compare(KeyOfValue()(v), key(x)); // v 键值小于目前节点的键值？
        x = comp ? left(x) : right(x);	// 遇「大」往左，遇「小于或等于」往右
      }
      //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指向安插点之父节点 y
      if (comp)	//如果离开while循环时comp为真，表示 父节点键值>v ，将安插在左孩子处
        if (j == begin())   // 如果j是最左节点
          return pair<iterator,bool>(__insert(x, y, v), true);
          // 以上，x 为安插点，y 为安插点之父节点，v 为新值。
        else	// 否则（安插点之父节点不是最左节点）
          --j;	// 调整 j，回头准备测试...
      if (key_compare(key(j.node), KeyOfValue()(v)))	
        // 小于新值（表示遇「小」，将安插于右侧）
        return pair<iterator,bool>(__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.node == header->left) // begin()
        if (size() > 0 && key_compare(KeyOfValue()(v), key(position.node)))
          return __insert(position.node, position.node, v);
      // first argument just needs to be non-null 
        else
          return insert_unique(v).first;
      else if (position.node == header) // end()
        if (key_compare(key(rightmost()), KeyOfValue()(v)))
          return __insert(0, rightmost(), v);
        else
          return insert_unique(v).first;
      else {
        iterator before = position;
        --before;
        if (key_compare(key(before.node), KeyOfValue()(v))
            && key_compare(KeyOfValue()(v), key(position.node)))
          if (right(before.node) == 0)
            return __insert(0, before.node, v); 
          else
            return __insert(position.node, position.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.node == header->left) // begin()
        if (size() > 0 && key_compare(KeyOfValue()(v), key(position.node)))
          return __insert(position.node, position.node, v);
      // first argument just needs to be non-null 
        else
          return insert_equal(v);
      else if (position.node == header) // end()
        if (!key_compare(KeyOfValue()(v), key(rightmost())))
          return __insert(0, rightmost(), v);
        else
          return insert_equal(v);
      else {
        iterator before = position;
        --before;
        if (!key_compare(KeyOfValue()(v), key(before.node))
            && !key_compare(key(position.node), KeyOfValue()(v)))
          if (right(before.node) == 0)
            return __insert(0, before.node, v); 
          else
            return __insert(position.node, position.node, v);
        // first argument just needs to be non-null 
        else
          return insert_equal(v);
      }
    }
    
    #ifdef __STL_MEMBER_TEMPLATES  
    
    template <class K, class V, class KoV, class Cmp, class Al> template<class II>
    void rb_tree<K, V, KoV, Cmp, Al>::insert_equal(II first, II last) {
      for ( ; first != last; ++first)
        insert_equal(*first);
    }
    
    template <class K, class V, class KoV, class Cmp, class Al> template<class II>
    void rb_tree<K, V, KoV, Cmp, Al>::insert_unique(II first, II last) {
      for ( ; first != last; ++first)
        insert_unique(*first);
    }
    
    #else /* __STL_MEMBER_TEMPLATES */
    
    template <class K, class V, class KoV, class Cmp, class Al>
    void
    rb_tree<K, V, KoV, Cmp, Al>::insert_equal(const V* first, const V* last) {
      for ( ; first != last; ++first)
        insert_equal(*first);
    }
    
    template <class K, class V, class KoV, class Cmp, class Al>
    void
    rb_tree<K, V, KoV, Cmp, Al>::insert_equal(const_iterator first,
                                              const_iterator last) {
      for ( ; first != last; ++first)
        insert_equal(*first);
    }
    
    template <class K, class V, class KoV, class Cmp, class A>
    void 
    rb_tree<K, V, KoV, Cmp, A>::insert_unique(const V* first, const V* last) {
      for ( ; first != last; ++first)
        insert_unique(*first);
    }
    
    template <class K, class V, class KoV, class Cmp, class A>
    void 
    rb_tree<K, V, KoV, Cmp, A>::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.node,
                                                              header->parent,
                                                              header->left,
                                                              header->right);
      destroy_node(y);
      --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;
    }
    //复制x到p
    template <class K, class V, class KeyOfValue, class Compare, class Alloc>
    typename rb_tree<K, V, KeyOfValue, Compare, Alloc>::link_type 
    rb_tree<K, V, KeyOfValue, Compare, Alloc>::__copy(link_type x, link_type p) {
                                    // structural copy.  x and p must be non-null.
      link_type top = clone_node(x);
      top->parent = p;
     
      __STL_TRY {
        if (x->right)
          top->right = __copy(right(x), top);
        p = top;
        x = left(x);
    
        while (x != 0) {
          link_type y = clone_node(x);
          p->left = y;
          y->parent = p;
          if (x->right)
            y->right = __copy(right(x), y);
          p = y;
          x = left(x);
        }
      }
      __STL_UNWIND(__erase(top));
    
      return top;
    }
    
    template <class Key, class Value, class KeyOfValue, class Compare, class Alloc>
    void rb_tree<Key, Value, KeyOfValue, Compare, Alloc>::__erase(link_type x) {
                                    // erase without rebalancing
      while (x != 0) {
        __erase(right(x));
        link_type y = 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++);
    }
    
    //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>::iterator 
    rb_tree<Key, Value, KeyOfValue, Compare, Alloc>::find(const Key& k) {
      link_type y = header;        // Last node which is not less than k. 
      link_type x = root();        // Current node. 
    
      while (x != 0) 
        // key_compare 是 function object。
        if (!key_compare(key(x), k)) 
          // 运行到这里，表示x键值大于k。遇到大值就向左走。
          y = x, x = left(x);	// 注意语法!逗号表达式
        else
          // 运行到这里，表示x键值小于k。遇到小值就向右走。
          x = right(x);
    
      iterator j = iterator(y);   
      return (j == end() || key_compare(k, key(j.node))) ? end() : j;
    }
    
    
    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 = header; /* Last node which is not less than k. */
      link_type x = root(); /* Current node. */
    
      while (x != 0) {
       
        if (!key_compare(key(x), k))
          y = x, x = left(x);	
        else
          x = right(x);
      }
      const_iterator j = const_iterator(y);   
      return (j == end() || key_compare(k, key(j.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 = header; /* Last node which is not less than k. */
      link_type x = root(); /* Current node. */
    
      while (x != 0) 
        if (!key_compare(key(x), k))
          y = x, x = left(x);
        else
          x = 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 = header; /* Last node which is not less than k. */
      link_type x = root(); /* Current node. */
    
      while (x != 0) 
        if (!key_compare(key(x), k))
          y = x, x = left(x);
        else
          x = 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 = header; /* Last node which is greater than k. */
      link_type x = root(); /* Current node. */
    
       while (x != 0) 
         if (key_compare(k, key(x)))
           y = x, x = left(x);
         else
           x = 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 = header; /* Last node which is greater than k. */
      link_type x = root(); /* Current node. */
    
       while (x != 0) 
         if (key_compare(k, key(x)))
           y = x, x = left(x);
         else
           x = 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->color == __rb_tree_black ? 1 : 0;
        if (node == root)
          return bc;
        else
          return bc + __black_count(node->parent, root); // 累加
      }
    }
    
    //验证己身这棵树是否符合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，符合RB树标准
      if (node_count == 0 || begin() == end())
        return node_count == 0 && begin() == end() &&
          header->left == header && header->right == header;
    
      //最左（叶）节点至 root 路径的黑节点个数
      int len = __black_count(leftmost(), root()); 
      //一下走访整个RB树，针对每个节点（从最小奥最大）……
      for (const_iterator it = begin(); it != end(); ++it) { 
        link_type x = (link_type) it.node; // __rb_tree_base_iterator::node
        link_type L = left(x);		// 这是左子节点
        link_type R = right(x); 	// 这是右子节点
    
        if (x->color == __rb_tree_red)
          if ((L && L->color == __rb_tree_red) ||
              (R && R->color == __rb_tree_red))
            return false;	// 父子节点同为红色，不合符RB树要求
    
        if (L && key_compare(key(x), key(L))) // 当前节点的键值小于左孩子节点的键值
          return false;         	// 不符合二叉查找树的要求
        if (R && key_compare(key(R), key(x))) // 当前节点的键值大于右孩子节点的键值
          return false;		// 不符合二叉查找树的要求
    
    	//[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
        if (!L && !R && __black_count(x, root()) != len) 
          return false;
      }
    
      if (leftmost() != __rb_tree_node_base::minimum(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 (rightmost() != __rb_tree_node_base::maximum(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;
    }
    
    __STL_END_NAMESPACE 
    
    #endif /* __SGI_STL_INTERNAL_TREE_H */
    
    // Local Variables:
    // mode:C++
    // End:

[Link 1]: http://blog.csdn.net/u013074465/article/details/42433461
[Center]: /images/20220805/7b379931e8064ee695897ec5ed936190.png
[Link 2]: http://blog.csdn.net/u013074465/article/details/41699891
[20150330201842035]: /images/20220805/3180d8b4f8eb4e31ac183461df1d0e96.png