• AVL树

AVL树又称为高度平衡的二叉搜索树，是1962年有俄罗斯的数学家G.M.Adel’son-Vel’skii和E.M.Landis提出来的。它能保持二叉树的高度平衡，尽量降低二叉树的高度，减少树的平均搜索长度。

• AVL树的性质

1. 左子树和右子树的高度之差的绝对值不超过1
2. 树中的每个左子树和右子树都是AVL树
3. 每个节点都有一个平衡因子（balance factor bf）,任一节点的平衡因子是1,0，-1.（每个节点的平衡因子等于右子树的高度减去左子树的高度）                                                                                                                                

• AVL树的效率

一棵AVL树有N个节点，其高度可以保持在log2N，插入/删除/查找的时间复杂度也是log2N。 （ps：log2N是表示log以2为底N的对数，evernote不支持公式。）

//AVLTree.h

#pragma once

template<class K,class V>
struct AVLTreeNode
{
	AVLTreeNode<K,V>* _left;
	AVLTreeNode<K,V>* _right;
	AVLTreeNode<K,V>* _parent;

	K _key;
	V _value;
	int _bf;

	AVLTreeNode(const K& key,const V& value)
		:_left(NULL)
		,_right(NULL)
		,_parent(NULL)
		,_key(key)
		,_value(value)
		,_bf(0)
	{}
};

template<class K,class V>
class AVLTree
{
	typedef AVLTreeNode<K,V> Node;
public:
	AVLTree()
		:_root(NULL)
	{}

	bool Insert(const K& key,const V& value)
	{
		//插入节点
		if(_root == NULL)
		{
			_root = new Node(key,value);
		}
		Node* cur = _root;
		Node* parent = NULL;
		while(cur)
		{
			if(cur->_key < key)
			{
				parent = cur;
				cur = cur->_right;
			}
			else if(cur->_key > key)
			{
				parent = cur;
				cur = cur->_left;
			}
			else
				return false;
		}
		cur = new Node(key,value);
		if(parent->_key < key)
		{
			parent->_right = cur;
			cur->_parent = parent;
		}
		else
		{
			parent->_left = cur;
			cur->_parent = parent;
		}

		//更新平衡因子
		while(parent)
		{
			//右孩子平衡因子加1，左孩子平衡因子减1
			if(parent->_right == cur)
				parent->_bf ++;
			else if(parent->_left == cur)
				parent->_bf --;
	
			if(parent->_bf == 0)
				break;
			else if(parent->_bf == 1 || parent->_bf == -1)
			{
				cur = parent;
				parent = cur->_parent;
			}
			else	//_bf = 2 || -2
			{
				if(parent->_bf == 2)	//parent->_bf == 2
				{
					if(cur->_bf == 1)
						RotateL(parent);
					else	//_bf = -1
						RotateRL(parent);
				}
				else if(parent->_bf == -2)	//parent->_bf == -2
				{
					if(cur->_bf == -1)
						RotateR(parent);
					else	//_bf = 1
						RotateLR(parent);
				}
				return true;
			}
		}
		return true;
	}
	void RotateL(Node* parent)
	{
		Node* subR = parent->_right;
		Node* subRL = subR->_left;
		parent->_right = subRL;
		if(subRL)
		{
			subRL->_parent = parent;
		}
		subR->_left = parent;
		Node* ppNode = parent->_parent;
		parent->_parent = subR;

		if(ppNode == NULL)
		{
			_root = subR;
			subR->_parent = NULL;
		}
		else
		{
			if(ppNode->_left == parent)
			{
				ppNode->_left = subR;
				subR->_parent = ppNode;
			}
			else
			{
				ppNode->_right = subR;
				subR->_parent = ppNode;
			}
		}
		subR->_bf = parent->_bf = 0;
	}
	void RotateR(Node* parent)
	{
		Node* subL = parent->_left;
		Node* subLR = subL->_right;
		parent->_left = subLR;
		if(subLR)
		{
			subLR->_parent = parent;
		}
		subL->_right = parent;

		Node* ppNode = parent->_parent;
		parent->_parent = subL;
		
		//判断parent是否有父亲节点
		if(ppNode == NULL)
		{
			_root = subL;
			subL->_parent = NULL;
		}
		else
		{
			if(ppNode->_left == parent)
			{
				ppNode->_left = subL;
				subL->_parent = ppNode;
			}
			else
			{
				ppNode->_right = subL;
				subL->_parent = ppNode;
			}
		}
		subL->_bf = parent->_bf = 0;
	}
	void RotateRL(Node* parent)
	{
		Node* subR = parent->_right;
		Node* subRL = subR->_left;
		int bf = subRL->_bf;

		RotateR(parent->_right);
		RotateL(parent);

		if(bf == 1)
		{
			subR->_bf = 0;
			parent->_bf = -1;
		}
		else if(bf == -1)
		{
			parent->_bf = 0;
			subR->_bf = 1;
		}
		else
		{
			subR->_bf = parent->_bf = 0;
		}
		subRL->_bf = 0;
	}
	void RotateLR(Node* parent)
	{
		Node* subL = parent->_left;
		Node* subLR = subL->_right;
		int bf = subLR->_bf;

		RotateL(parent->_left);
		RotateR(parent);

		if(bf == 1)
		{
			subL->_bf = -1;
			parent->_bf = 0;
		}
		else if(bf == -1)
		{
			parent->_bf = 1;
			subL->_bf = 0;
		}
		else 
		{
			subL->_bf = parent->_bf = 0;
		}
		subLR->_bf = 0;
	}

	void InOrder()
	{
		_InOrder(_root);
		cout<<endl;
	}

	int Height(Node* root)
	{
		if(root == NULL)
		{
			return 0;
		}
		int left = Height(root->_left);
		int right = Height(root->_right);

		return left > right ? (left+1) : (right+1); 
	}

	bool IsBlance()
	{
		return _IsBlance(_root);
	}

	bool _IsBlance(Node* root)
	{
		if(root == NULL)
		{
			return true;
		}
		int left = Height(root->_left);
		int right = Height(root->_right);

		if((right-left) != root->_bf)
		{
			cout<<"平衡因子异常"<<root->_key<<endl;
			return false;
		}
		return abs(right-left < 2)
			&& _IsBlance(root->_left)
			&& _IsBlance(root->_right);
	}

	void _InOrder(Node* root)
	{
		if(root == NULL)
		{
			return;
		}
		_InOrder(root->_left);
		cout<<root->_key<<" ";
		_InOrder(root->_right);
	}

protected:
	Node* _root;
};

void TestTree()
{
	AVLTree<int,int> tree;
	int a[]={16, 3, 7, 11, 9, 26, 18, 14, 15};
	for(size_t i = 0; i < sizeof(a)/sizeof(a[0]); ++i)
	{
		tree.Insert(a[i],i);
	}
	tree.InOrder();
	cout<<"Blance?"<<tree.IsBlance()<<endl;
	getchar();
}

void TestTree1()
{
	AVLTree<int,int> tree1;
	int array1[]={4, 2, 6, 1, 3, 5, 15, 7, 16, 14};
	for(size_t i = 0; i < sizeof(array1)/sizeof(array1[0]); ++i)
	{
		tree1.Insert(array1[i],i);
	}
	tree1.InOrder();
	cout<<"Blance?"<<tree1.IsBlance()<<endl;
	getchar();
}

//Test.cpp

#include<iostream>
using namespace std;

#include"AVLTree.h"

int main()
{
	//TestTree();
	TestTree1();
	getchar();
	return 0;
}