红黑树详解_C/C++/C#

红黑树红黑树的概念红黑树，是一种二叉搜索树，但在每个结点上增加一个存储位表示结点的颜色，可以是 Red 或 Black 。通过对任何一条从根到叶子的路径上各个结点着色方式的限制，红黑树确保没有一条路径会比其他路径长出俩倍，因而是接近平衡的。红黑树的性质 1. 每个结点不是红色就是黑色 2. 根节点是黑色的 3. 如果一个节点是红色的，则它的两个孩子结点是黑色的 4. 对于每个结点，从该结点到其所有后代叶结点的简单路径上，均包含相同数目的黑色结点 5. 每个叶子结点都是黑色的 ( 此处的叶子结点指的是空结点 ) 思考：为什么满足上面的性质，红黑树就能保证：其最长路径中节点个数不会超过最短路径节点个数的两倍？答：假设一条路径黑色节点为H，那么多短路径就是H,最长路径为2H,大佬就是大佬，也不知怎么想出来的。红黑树节点的定义

enum Colour
{
	RED,
	BLACK
};
template
class RBTreeNode
{
public:
	RBTreeNode(const std::pair& kv)
		:_left(nullptr)
		,_right(nullptr)
		,_parent(nullptr)
		,_kv(kv)
		,_col(RED)
	{

	}
	RBTreeNode* _left;
	RBTreeNode* _right;
	RBTreeNode* _parent;
	std::pair _kv;
	Colour _col;
};

思考：在节点的定义中，为什么要将节点的默认颜色给成红色的？答：节点定义成红色是为了方便处理。红黑树结构为了后续实现关联式容器简单，红黑树的实现中增加一个头结点，因为跟节点必须为黑色，为了与根节点进行区分，将头结点给成黑色，并且让头结点的 Parent 域指向红黑树的根节点， Left 域指向红黑树中最小的节点，Right 域指向红黑树中最大的节点，如下：红黑树的插入操作红黑树是在二叉搜索树的基础上加上其平衡限制条件，因此红黑树的插入可分为两步： 1. 按照二叉搜索的树规则插入新节点

template
class RBTree
{
public:
	typedef RBTreeNode Node;
	RBTree()
		:_root(nullptr)
	{

	}
	bool insert(const std::pair &kv)
	{
		if (_root == nullptr)
		{
			_root = new Node(kv);
			_root->_col = BLACK;
			return true;
		}
		Node* parent = nullptr;
		Node* cur = _root;
		while (cur != nullptr)
		{
			if (kv.first > cur->_kv.first)
			{
				parent = cur;
				cur = cur->_right;
			}
			else if (kv.first < cur->_kv.first)
			{
				parent = cur;
				cur = cur->_left;
			}
			else
			{
				return false;
			}
		}
		//新增节点
		cur = new Node(kv);
		cur->_col = RED;
		if (kv.first > parent->_kv.first)
		{
			parent->_right = cur;
			cur->_parent = parent;
		}
		else
		{
			parent->_left = cur;
			cur->_parent = parent;
		}
}

2. 检测新节点插入后，红黑树的性质是否造到破坏因为新节点的默认颜色是红色，因此：如果其双亲节点的颜色是黑色，没有违反红黑树任何性质，则不需要调整；但当新插入节点的双亲节点颜色为红色时，就违反了性质三不能有连在一起的红色节点，此时需要对红黑树分情况来讨论：约定 :cur 为当前节点， p 为父节点， g 为祖父节点， u 为叔叔节点情况一: cur为红，p为红，g为黑，u存在且为红解决方式：将p,u改为黑，g改为红，然后把g当成cur，继续向上调整情况二: cur为红，p为红，g为黑，u不存在/u为黑

说明：u的情况右两种

1.如果u节点不存在，则cur一定是新插入的节点，如果cur不是新插入节点，则cur和p一定有一个节点的颜色是黑色，就不满足性质4，每条路径黑色节点的个数相同。

2.如果u节点存在，则其一定是黑色，那么cur节点原来的颜色也一定是黑色的，现在看到其是红色的原因因为cur的子树在调整过程中将cur的颜色由黑色变成红色的。

p为g的左孩子，cur为p的左孩子，则进行右单旋

p为g的右孩子，cur为p的右孩子，则进行左单旋

p,g变色 --p变黑，g变红。

情况3.cur为红，p为红，g为黑，u不存在/u为黑

p 为 g 的左孩子， cur 为 p 的右孩子，则针对 p 做左单旋转；相反， p 为 g 的右孩子， cur 为 p 的左孩子，则针对 p 做右单旋转则转换成了情况 2

//控制平衡
		while (parent != nullptr && parent->_col == RED)  //如果父亲存在并且父亲的节点为红色
		{
			Node* grandfather = parent->_parent;
			if (parent == grandfather->_left)
			{
				Node* uncle = grandfather->_right;
				//情况1 ：叔叔节点存在且叔叔节点颜色为红
				if (uncle != nullptr && uncle->_col == RED)
				{
					parent->_col = uncle->_col = BLACK;  //父亲节点和叔叔节点变黑
					grandfather->_col = RED;
					
					//grandfather可能不是根节点，继续往上更新
					cur = grandfather;
					parent = cur->_parent;
				}
				else //情况2 + 3，uncle不存在 / 存在且为黑    可能是由情况1 -> 情况2 / 情况 3
				{
					//        g
					//      p
					//    c
					if (cur == parent->_left)   
					{
						//右单旋
						RotateR(grandfather);
						parent->_col = BLACK;
						grandfather->_col = RED;
					}
					else
					{
						//触发左右双旋
						RotateL(parent);
						RotateR(grandfather);
						cur->_col = BLACK;
						grandfather->_col = RED;
					}
					break;
				}
			}
			else
			{
				Node* uncle = grandfather->_left;
				//情况1： uncle 存在且 uncle的颜色为红
				if (uncle != nullptr && uncle->_col == RED)
				{
					parent->_col = uncle->_col = BLACK;  //叔叔和父亲变色
					grandfather->_col = RED;

					//grandfather可能不是根节点，所以可能一直往上更新
					cur = grandfather;
					parent = cur->_parent;
				}
				else  //情况2 + 3 ，uncle不存在 / 存在且为黑
				{
					if (cur == parent->_right)  //触发左旋
					{
						RotateL(grandfather);
						parent->_col = BLACK;
						grandfather->_col = RED;
					}
					else  
					{
						//触发右左双旋
						RotateR(parent);
						RotateL(grandfather);
						grandfather->_col = RED;
						cur->_col = BLACK;
					}
					break;
				}
			}
		}
		_root->_col = BLACK;
		return true;
	}

旋转代码：

void RotateR(Node* parent)
	{
		Node* subL = parent->_left;
		Node* subLR = subL->_right;
		Node* parentparent = parent->_parent;

		parent->_left = subLR;
		if (subLR != nullptr)
		{
			subLR->_parent = parent;
		}
		subL->_right = parent;
		parent->_parent = subL;
		if (_root == parent)
		{
			_root = subL;
			subL->_parent = nullptr;
		}
		else
		{
			if (parentparent->_left == parent)
				parentparent->_left = subL;
			else
				parentparent->_right = subL;
			subL->_parent = parentparent;
		}
	}
	void RotateL(Node* parent)
	{
		
		Node* subR = parent->_right;
		Node* subRL = subR->_left;
		Node* parentparent = parent->_parent;

		parent->_right = subRL;
		if (subRL != nullptr)
			subRL->_parent = parent;

		subR->_left = parent;
		parent->_parent = subR;
		if (parent == _root)
		{
			_root = subR;
			subR->_parent = nullptr;
		}
		else
		{
			if (parentparent->_left == parent)
				parentparent->_left = subR;
			else
				parentparent->_right = subR;

			subR->_parent = parentparent;
		}
	}

红黑树的验证红黑树的检测分为两步： 1. 检测其是否满足二叉搜索树 ( 中序遍历是否为有序序列 ) 2. 检测其是否满足红黑树的性质

bool IsBalance()
	{
		if (_root && _root->_col == RED)
		{
			std::cout << "根节点不是黑色" << std::endl;
			return false;
		}

		// 最左路径黑色节点数量做基准值
		int banchmark = 0;
		Node* left = _root;
		while (left)
		{
			if (left->_col == BLACK)
				++banchmark;

			left = left->_left;
		}

		int blackNum = 0;
		return _IsBalance(_root, banchmark, blackNum);
	}

	bool _IsBalance(Node* root, int banchmark, int blackNum)
	{
		if (root == nullptr)
		{
			if (banchmark != blackNum)
			{
				std::cout << "存在路径黑色节点的数量不相等" << std::endl;
				return false;
			}

			return true;
		}

		if (root->_col == RED && root->_parent->_col == RED)
		{
			std::cout << "出现连续红色节点" << std::endl;
			return false;
		}

		if (root->_col == BLACK)
		{
			++blackNum;
		}

		return _IsBalance(root->_left, banchmark, blackNum)
			&& _IsBalance(root->_right, banchmark, blackNum);
	}

红黑树的删除红黑树的删除本节不做讲解，有兴趣的同学可参考：《算法导论》或者《 STL 源码剖析》 http://www.cnblogs.com/fornever/archive/2011/12/02/2270692.html http://blog.csdn.net/chenhuajie123/article/details/11951777 红黑树与AVL树的比较红黑树和 AVL 树都是高效的平衡二叉树，增删改查的时间复杂度都是 O(log2 N ) ，红黑树不追求绝对平衡，其只需保证最长路径不超过最短路径的 2 倍，相对而言，降低了插入和旋转的次数，所以在经常进行增删的结构中性能比 AVL 树更优，而且红黑树实现比较简单，所以实际运用中红黑树更多。红黑树的应用 1. C++ STL 库 -- map/set 、 mutil_map/mutil_set 2. Java 库 3. linux 内核 4. 其他一些库 http://www.cnblogs.com/yangecnu/p/Introduce-Red-Black-Tree.html 红黑树全部代码： RBTree.h

#pragma once
#include
enum Colour
{
	RED,
	BLACK
};
template
class RBTreeNode
{
public:
	RBTreeNode(const std::pair& kv)
		:_left(nullptr)
		,_right(nullptr)
		,_parent(nullptr)
		,_kv(kv)
		,_col(RED)
	{

	}
	RBTreeNode* _left;
	RBTreeNode* _right;
	RBTreeNode* _parent;
	std::pair _kv;
	Colour _col;
};

template
class RBTree
{
public:
	typedef RBTreeNode Node;
	RBTree()
		:_root(nullptr)
	{

	}
	bool insert(const std::pair &kv)
	{
		if (_root == nullptr)
		{
			_root = new Node(kv);
			_root->_col = BLACK;
			return true;
		}
		Node* parent = nullptr;
		Node* cur = _root;
		while (cur != nullptr)
		{
			if (kv.first > cur->_kv.first)
			{
				parent = cur;
				cur = cur->_right;
			}
			else if (kv.first < cur->_kv.first)
			{
				parent = cur;
				cur = cur->_left;
			}
			else
			{
				return false;
			}
		}
		//新增节点
		cur = new Node(kv);
		cur->_col = RED;
		if (kv.first > parent->_kv.first)
		{
			parent->_right = cur;
			cur->_parent = parent;
		}
		else
		{
			parent->_left = cur;
			cur->_parent = parent;
		}
		//控制平衡
		while (parent != nullptr && parent->_col == RED)  //如果父亲存在并且父亲的节点为红色
		{
			Node* grandfather = parent->_parent;
			if (parent == grandfather->_left)
			{
				Node* uncle = grandfather->_right;
				//情况1 ：叔叔节点存在且叔叔节点颜色为红
				if (uncle != nullptr && uncle->_col == RED)
				{
					parent->_col = uncle->_col = BLACK;  //父亲节点和叔叔节点变黑
					grandfather->_col = RED;
					
					//grandfather可能不是根节点，继续往上更新
					cur = grandfather;
					parent = cur->_parent;
				}
				else //情况2 + 3，uncle不存在 / 存在且为黑    可能是由情况1 -> 情况2 / 情况 3
				{
					//        g
					//      p
					//    c
					if (cur == parent->_left)   
					{
						//右单旋
						RotateR(grandfather);
						parent->_col = BLACK;
						grandfather->_col = RED;
					}
					else
					{
						//触发左右双旋
						RotateL(parent);
						RotateR(grandfather);
						cur->_col = BLACK;
						grandfather->_col = RED;
					}
					break;
				}
			}
			else
			{
				Node* uncle = grandfather->_left;
				//情况1： uncle 存在且 uncle的颜色为红
				if (uncle != nullptr && uncle->_col == RED)
				{
					parent->_col = uncle->_col = BLACK;  //叔叔和父亲变色
					grandfather->_col = RED;

					//grandfather可能不是根节点，所以可能一直往上更新
					cur = grandfather;
					parent = cur->_parent;
				}
				else  //情况2 + 3 ，uncle不存在 / 存在且为黑
				{
					if (cur == parent->_right)  //触发左旋
					{
						RotateL(grandfather);
						parent->_col = BLACK;
						grandfather->_col = RED;
					}
					else  
					{
						//触发右左双旋
						RotateR(parent);
						RotateL(grandfather);
						grandfather->_col = RED;
						cur->_col = BLACK;
					}
					break;
				}
			}
		}
		_root->_col = BLACK;
		return true;
	}
private:
	void RotateR(Node* parent)
	{
		Node* subL = parent->_left;
		Node* subLR = subL->_right;
		Node* parentparent = parent->_parent;

		parent->_left = subLR;
		if (subLR != nullptr)
		{
			subLR->_parent = parent;
		}
		subL->_right = parent;
		parent->_parent = subL;
		if (_root == parent)
		{
			_root = subL;
			subL->_parent = nullptr;
		}
		else
		{
			if (parentparent->_left == parent)
				parentparent->_left = subL;
			else
				parentparent->_right = subL;
			subL->_parent = parentparent;
		}
	}
	void RotateL(Node* parent)
	{
		
		Node* subR = parent->_right;
		Node* subRL = subR->_left;
		Node* parentparent = parent->_parent;

		parent->_right = subRL;
		if (subRL != nullptr)
			subRL->_parent = parent;

		subR->_left = parent;
		parent->_parent = subR;
		if (parent == _root)
		{
			_root = subR;
			subR->_parent = nullptr;
		}
		else
		{
			if (parentparent->_left == parent)
				parentparent->_left = subR;
			else
				parentparent->_right = subR;

			subR->_parent = parentparent;
		}
	}
public:
	void inoder()
	{
		Node* cur = _root;
		Node* MostRight = nullptr;
		while (cur != nullptr)
		{
			MostRight = cur->_left;
			if (MostRight != nullptr)
			{
				while (MostRight->_right != nullptr && MostRight->_right != cur)
				{
					MostRight = MostRight->_right;
				}
				if (MostRight->_right == nullptr)
				{
					MostRight->_right = cur;
					cur = cur->_left;
					continue;
				}
				else
				{
					MostRight->_right = nullptr;
				}
			}
			std::cout << cur->_kv.first << ":" << cur->_kv.second << std::endl;
			cur = cur->_right;
		}
	}
	bool IsBalance()
	{
		if (_root && _root->_col == RED)
		{
			std::cout << "根节点不是黑色" << std::endl;
			return false;
		}

		// 最左路径黑色节点数量做基准值
		int banchmark = 0;
		Node* left = _root;
		while (left)
		{
			if (left->_col == BLACK)
				++banchmark;

			left = left->_left;
		}

		int blackNum = 0;
		return _IsBalance(_root, banchmark, blackNum);
	}

	bool _IsBalance(Node* root, int banchmark, int blackNum)
	{
		if (root == nullptr)
		{
			if (banchmark != blackNum)
			{
				std::cout << "存在路径黑色节点的数量不相等" << std::endl;
				return false;
			}

			return true;
		}

		if (root->_col == RED && root->_parent->_col == RED)
		{
			std::cout << "出现连续红色节点" << std::endl;
			return false;
		}

		if (root->_col == BLACK)
		{
			++blackNum;
		}

		return _IsBalance(root->_left, banchmark, blackNum)
			&& _IsBalance(root->_right, banchmark, blackNum);
	}
private:
	Node* _root;
};

test.cpp

#include"AVL.h"
#include"RBTree.h"

void AVLTreeTest(void)
{
	AVLTree av;
	int array[] = { 9,6,4,3,7,3,2,0,544,43224,23,423,42,4234,2,4242,423,42,7,11,98,5 };
	for (const auto& e : array)
	{
		av.insert(std::pair(e,e));
	}
	av.inoder();
}
void RBTreeTest(void)
{
	RBTree rb;
	int array[] = { 9,6,4,3,7,3,2,0,544,43224,23,423,42,4234,2,4242,423,42,7,11,98,5 };
	for (const auto& e : array)
	{
		rb.insert(std::pair(e, e));
		std::cout << rb.IsBalance() << std::endl;
	}
	rb.inoder();
}
int main()
{
	//AVLTreeTest();
	RBTreeTest();
}

红黑树详解

C/C++/C#相关栏目本月热门文章