AVL树基于二叉搜索树，为了解决快速查找出现次数最多的数，引入了键值对key value

二叉搜索树参考：点击打开链接

1.0 AVL树定义：

2.0 思路：

注： 红圈为插入的节点

3.0 代码：

AVL.h

#pragma once
#include<iostream>
using namespace std;
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)   //平衡因子
	{}
};
template<class k, class v>
class AVLtree
{
	typedef AVLtreeNode<k, v> Node;
	typedef AVLtreeNode<k, v>* PNode;
public:
	AVLtree()
		:_root(NULL)
	{}
	bool Insert(const k& key, const v& value)
	{
		return _Insert(_root, key, value);
	}
	void InOrderReverse()
	{
		return _InOrderReverse(_root);
	}
	void showbf()
	{
		_showbf(_root);
	}
	bool IsBalanceTree()
	{
		return _IsBalanceTree(_root);
	}
protected:
	void _rotateL(PNode& parent)
	{
		PNode PSubR = parent->_right;
		PNode PSubRL = PSubR->_left;
		parent->_right = PSubRL;
		if (PSubRL)
		{
			PSubRL->_parent = parent;
		}
		PNode pparent = parent->_parent; //记录parent的双亲节点必须在更改它的双亲节点之前

		PSubR->_left = parent;
		parent->_parent = PSubR;
		PSubR->_parent = pparent;
		if (NULL == pparent)
		{
			_root = PSubR;
			PSubR->_parent = NULL;
		}
		else
		{
			if (pparent->_left == parent)
				pparent->_left = PSubR;
			else
				pparent->_right = PSubR;
		}
		//更新平衡因子
		PSubR->_bf= parent->_bf = 0;
	}
	void _rotateR(PNode& parent)
	{
		PNode PSubL = parent->_left;
		PNode PSubLR = PSubL->_right;

		parent->_left = PSubLR;
		if (PSubLR)
			PSubLR->_parent = parent;
		PSubL->_right = parent;
		PNode pparent = parent->_parent;  //记录parent的双亲节点必须在更改它的双亲节点之前
		parent->_parent = PSubL;
		PSubL->_parent = pparent;
		if (NULL == pparent)
		{
			_root = PSubL;
			PSubL->_parent = NULL;
		}
		else
		{
			if (pparent->_left == parent)               //pparent记录着parent原来的左右孩子
				pparent->_left = PSubL;
			else
				pparent->_right = PSubL;
		}
		//更新平衡因子
		PSubL->_bf = parent->_bf = 0;
	}
	void _rotateLR(PNode& parent)
	{
		PNode PSubL = parent->_left;
		PNode PSubLR = PSubL->_right;
		int bf = PSubLR->_bf;
		_rotateL(parent->_left);
		_rotateR(parent);
		//更新平衡因子
		if (bf == 1)
			PSubL->_bf = -1;
		else if (bf == -1)
			parent->_bf = 1;
		else
			parent->_bf = PSubL->_bf = 0;
	}
	void _rotateRL(PNode& parent)
	{
		PNode PSubR = parent->_right;
		PNode PSubRL= PSubR->_left;
		int bf = PSubRL->_bf;
		_rotateR(parent->_right);
		_rotateL(parent);
		//更新平衡因子
		if (bf == 1)
			parent->_bf = -1;
		else if (bf == -1)
			PSubR->_bf = 1;
		else
			parent->_bf = PSubR->_bf = 0;
	}

	bool _Insert(PNode& root, const k& key, const v& value)
	{
		if (NULL == root)   //树为空
		{
			_root = new Node(key, value);
			return true;
		}
	//找待插入节点的位置
		PNode PCur = root;
		PNode parent = NULL;
		while (PCur)
		{
			if (key > PCur->_key)
			{
				parent = PCur;
				PCur = PCur->_right;
			}
			else if (key < PCur->_key)
			{
				parent = PCur;
				PCur = PCur->_left;
			}
			else
				return false;
		}
		//插入节点
		PCur = new Node(key, value);
		if (key>parent->_key)
			parent->_right = PCur;
		else
			parent->_left = PCur;
		PCur->_parent = parent;
		//更新平衡因子
		while (parent)
		{
			if (parent->_left ==PCur)
				parent->_bf--;
			else
				parent->_bf++;
			if (parent->_bf == 0)
				return true;
			else if (parent->_bf == -1 || parent->_bf == 1)
			{
				PCur = parent;
				parent = PCur->_parent;
			}
			else      //parent->_bf == -2 || parent->_nf == 2
			{
				if (parent->_bf == 2)     
				{
					if (PCur->_bf == 1)
						_rotateL(parent);
					else
						_rotateRL(parent);
				}
				else                     //parent->_bf == -2
				{
					if (PCur->_bf == -1)
					{
						_rotateR(parent);
					}
					else
						_rotateLR(parent);
				}
				break;
			}
		}
		return true;
	}
	void _InOrderReverse(PNode root)
	{
		if (root)
		{
			_InOrderReverse(root->_left);
			cout << root->_key << " ";
			_InOrderReverse(root->_right);
		}
	}
	//打印节点平衡因子（中序）
	void _showbf(PNode root)
	{
		if (root)
		{
			_showbf(root->_left);
			cout << root->_bf << " ";
			_showbf(root->_right);
		}
	}
	/*bool _IsBalanceTree(PNode root)
	{

		if (NULL == root)
			return true;
			if (!(root->_bf >= -1 && root->_bf <= 1))
				return false;
			return _IsBalanceTree(root->_left) && _IsBalanceTree(root->_right);
	}*/
	//求以该节点为根的树的高度
	size_t _Height(PNode root)
	{
		if (NULL == root)
			return 0;
		size_t LeftHeight = _Height(root->_left);
		size_t RightHeight = _Height(root->_right);
		return (LeftHeight > RightHeight ? LeftHeight + 1 : RightHeight + 1);
	}
	bool _IsBalanceTree(PNode root)
	{

		if (NULL == root)
			return true;
		size_t LeftHeight = _Height(root->_left);
		size_t RightHeight = _Height(root->_right);
		int distance = LeftHeight - LeftHeight;
		if (abs(distance)>1&&root->_bf!=distance)
			return false;
		return _IsBalanceTree(root->_left) && _IsBalanceTree(root->_right);
	}
private:
	PNode _root;
};
void AVL_Test()
{
	//int arr[] = { 4, 2, 6, 1, 3, 5, 15, 7, 16, 14 };
	int arr[] = { 2,8,5,7,4,63,31};
	int size = sizeof(arr) / sizeof(arr[0]);
	AVLtree<int, int> at;
	for (int i = 0; i < size; ++i)
		at.Insert(arr[i], i);
	cout <<"中序遍历结果："<< endl;
	at.InOrderReverse();
	cout << endl;
	cout<< at.IsBalanceTree() << endl;
	at.showbf();
	cout << endl;
} 

main.cpp

#define _CRT_SECURE_NO_WARNINGS 1
#pragma once
#include"AVL.h"
#include<iostream>
using namespace std;
int main()
{
	AVL_Test();
	system("pause");
	return 0;
}

4.0 结果：