【C++】——AVL树-优快云博客

本文链接：https://blog.youkuaiyun.com/super_coders/article/details/142792566

文章目录

AVL树是什么
AVL树的实现
完整代码

AVL树是什么

AVL树是在平衡二叉树的基础上改进的。

在这里插入图片描述

特点：

AVL树每个节点的平衡因子只可能是 -1, 1, 0
在AVL树中，任何节点的两个子树的高度最大差别为1，因此它也被称为高度平衡树。
AVL树在插入和删除节点时，能够通过旋转操作自动调整树的结构，以保持树的平衡性。

AVL树的实现

功能：

插入
删除
查找

掌握插入即可

AVL树的节点

和二叉搜索树相似，AVL树多了一个平衡因子

template<class K, class V>
struct AVLTreeNode
{
	// 需要parent指针，后续更新平衡因子可以看到
	pair<K, V> _kv;
	AVLTreeNode<K, V>* _left;
	AVLTreeNode<K, V>* _right;
	AVLTreeNode<K, V>* _parent;

	int _bf; // balance factor

	AVLTreeNode(const pair<K, V>& kv)
		:_kv(kv)
		, _left(nullptr)
		, _right(nullptr)
		, _parent(nullptr)
		, _bf(0)
	{}
};

ALV树的插入

插入数据要更新平衡因子，如果高度差大于一，就需要通过旋转平衡。

AVL树旋转操作分4种类型：
右单旋
左单旋
右左双旋
左右双旋

右单旋

在这里插入图片描述
2. 左单旋

在这里插入图片描述
3. 左右双旋

在这里插入图片描述

右左双旋
左右双旋反过来就是右左双旋。（省略画图）

bool Insert(const pair<K, V>& kv)    //插入节点
{
	if (_root == nullptr)
	{
		_root = new Node(kv);
		return true;
	}

	Node* parent = nullptr;
	Node* cur = _root;
	while (cur)
	{
		if (cur->_kv.first < kv.first)
		{
			parent = cur;
			cur = cur->_right;
		}
		else if (cur->_kv.first > kv.first)
		{
			parent = cur;
			cur = cur->_left;
		}
		else
		{
			return false;
		}
	}
	cur = new Node(kv);
	if (parent->_kv.first < kv.first)
	{
		parent->_right = cur;
		cur->_parent = parent;
	}
	else
	{
		parent->_left = cur;
		cur->_parent = parent;
	}

	//管控平衡
	while (parent)    //当为根时，停止
	{
		if (cur == parent->_left)
		{
			parent->_bf--;
		}
		else
		{
			parent->_bf++;
		}
		if (parent->_bf == 0)
		{
			break;
		}
		else if (parent->_bf == 1 || parent->_bf == -1)
		{
			cur = parent;
			parent = parent->_parent;
		}
		else if (parent->_bf == 2 || parent->_bf == -2) //旋转
		{
			if (parent->_bf == 2 && cur->_bf == 1)
			{
				//左单旋
				RotateL(parent);
				break;
			}
			else if (parent->_bf == -2 && cur->_bf == -1)
			{
				//右单旋
				RotateR(parent);
				break;
			}
			else if (parent->_bf == 2 && cur->_bf == -1)
			{
				//RL型
				RotateRL(parent);
				break;
			}
			else if (parent->_bf == -2 && cur->_bf == 1)
			{
				//LR型
				RotateLR(parent);
				break;
			}
		}
		else
		{
			assert(false);
		}
	}
	return true;
}

AVL树的旋转

// 右单旋
void RotateR(Node* parent)
{
	Node* subL = parent->_left;
	Node* subLR = subL->_right;
	if (subLR) // 防止野指针
		subLR->_parent = parent;
		
	Node* pParent = parent->_parent;
	subL->_right = parent;
	parent->_parent = subL;
	if (parent == _root)
	{
		_root = subL;
		subL->_parent = nullptr;
	}
	else
	{
		if (pParent->_left == parent)
		{
			pParent->_left = subL;
		}
		else
		{
			pParent->_right = subL;
		}

		subL->_parent = pParent;
	}
	subL->_bf = 0;
	parent->_bf = 0;
}

// 左单旋
void RotateL(Node* parent)
{
	Node* subR = parent->_right;
	Node* subRL = subL->_left;

	if (subRL) // 防止野指针
		parent->_right = subRL;

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

		suLR->_parent = pParent;
	}
	subR->_bf = 0;
	parent->_bf = 0;
}

// 左右双旋
void RotateLR(Node* parent)
{
	Node* subL = parent->_left;
	Node* subLR = subL->_right;
	int bf = subLR->_bf;

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

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


// 右左双旋
void RotateRL(Node* parent)
{
	Node* subR = parent->_right;
	Node* subRL = subR->_left;
	int bf = subRL->_bf;

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

	if (bf == -1)
	{
		subRL->_bf = 0;
		subR->_bf = 1;
		parent->_bf = 0;
	}
	else if (bf == 1)
	{
		subRL->_bf = 0;
		subR->_bf = 0;
		parent->_bf = -1;
	}
	else if(bf == 0)
	{
		subRL->_bf = 0;
		subR->_bf = 0;
		parent->_bf = 0;
	}
	else
	{
		perror("bf failed");
	}
}

AVL树打印与检查

void _Inorder(Node* root)
{
	if (root == nullptr)
		return;
	_Inorder(root->_left);
	cout << root->_kv.first << " ";
	_Inorder(root->_right);
}
int _High(Node* root)
{
	if (root == nullptr)
		return 0;

	int LeftHigh = _High(root->_left);
	int RightHigh = _High(root->_right);
	return LeftHigh > RightHigh ? LeftHigh + 1 : RightHigh + 1;
}
bool _IsBalance(Node* root)
{
	if (root == nullptr)
		return true;
	int LeftHigh = _High(root->_left);
	int RightHigh = _High(root->_right);
	if (RightHigh - LeftHigh != root->_bf)
	{
		cout << _root->_kv.first << "当前节点平衡因子有问题" << endl;
		return false;
	}
	return abs(LeftHigh - RightHigh) < 2
		&& _IsBalance(root->_left)
		&& _IsBalance(root->_right);
}

完整代码

头文件

#pragma once
#include<iostream>
#include<assert.h>
using namespace std;


template<class K, class V>
struct AVLTreeNode
{
	AVLTreeNode<K, V>* _left;    //左子树
	AVLTreeNode<K, V>* _right;   //右子树
	AVLTreeNode<K, V>* _parent;  //父亲
	pair<K, V> _kv;       //存放节点值的
	int _bf;    //平衡因子（通过这个可以直到左右子树存在情况）

	//构造函数
	AVLTreeNode(const pair<K, V>& kv)
		:_left(nullptr)
		, _right(nullptr)
		, _parent(nullptr)
		, _kv(kv)
		, _bf(0)     //平衡因子起始值是0，当左子树插入一个节点时-1，右子树插入一个节点时+1
	{}
};

template<class K, class V>
class AVLTree
{
	typedef AVLTreeNode<K, V> Node;
public:
	bool Insert(const pair<K, V>& kv)    //插入节点
	{
		if (_root == nullptr)
		{
			_root = new Node(kv);
			return true;
		}

		Node* parent = nullptr;
		Node* cur = _root;
		while (cur)
		{
			if (cur->_kv.first < kv.first)
			{
				parent = cur;
				cur = cur->_right;
			}
			else if (cur->_kv.first > kv.first)
			{
				parent = cur;
				cur = cur->_left;
			}
			else
			{
				return false;
			}
		}
		cur = new Node(kv);
		if (parent->_kv.first < kv.first)
		{
			parent->_right = cur;
			cur->_parent = parent;
		}
		else
		{
			parent->_left = cur;
			cur->_parent = parent;
		}

		//管控平衡
		while (parent)    //当为根时，停止
		{
			if (cur == parent->_left)
			{
				parent->_bf--;
			}
			else
			{
				parent->_bf++;
			}
			if (parent->_bf == 0)
			{
				break;
			}
			else if (parent->_bf == 1 || parent->_bf == -1)
			{
				cur = parent;
				parent = parent->_parent;
			}
			else if (parent->_bf == 2 || parent->_bf == -2) //旋转
			{
				if (parent->_bf == 2 && cur->_bf == 1)
				{
					//左单旋
					RotateL(parent);
					break;
				}
				else if (parent->_bf == -2 && cur->_bf == -1)
				{
					//右单旋
					RotateR(parent);
					break;
				}
				else if (parent->_bf == 2 && cur->_bf == -1)
				{
					//RL型
					RotateRL(parent);
					break;
				}
				else if (parent->_bf == -2 && cur->_bf == 1)
				{
					//LR型
					RotateLR(parent);
					break;
				}
			}
			else
			{
				assert(false);
			}
		}
		return true;
	}

	void RotateL(Node* parent)   //左单旋（RR型）
	{
		Node* subR = parent->_right;
		Node* subRL = subR->_left;
		Node* parentParent = parent->_parent;

		parent->_right = subRL;
		subR->_left = parent;

		if (subRL)
			subRL->_parent = parent;

		if (_root == parent)
		{
			_root = subR;
			subR->_parent = nullptr;
		}
		else
		{
			if (parentParent->_left == parent)
			{
				parentParent->_left = subR;
			}
			else
			{
				parentParent->_right = subR;
			}
			subR->_parent = parentParent;
		}
		parent->_parent = subR;
		parent->_bf = 0;
		subR->_bf = 0;
	}
	void RotateR(Node* parent)   //右单旋（LL型）
	{
		Node* subL = parent->_left;
		Node* subLR = subL->_right;
		Node* parentParent = parent->_parent;

		parent->_left = subLR;
		subL->_right = parent;

		if (subLR)
		{
			subLR->_parent = parent;
		}
		if (_root == parent)
		{
			_root = subL;
			subL->_parent = nullptr;
		}
		else
		{
			if (parentParent->_left == parent)
			{
				parentParent->_left = subL;
			}
			else
			{
				parentParent->_right = subL;
			}
			subL->_parent = parentParent;
		}
		parent->_parent = subL;
		subL->_bf = parent->_bf = 0;
	}
	void RotateRL(Node* parent)      //先右单旋，再左单旋（RL型）
	{
		Node* subR = parent->_right;
		Node* subRL = subR->_left;
		int bf = subRL->_bf;

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

		if (bf == 0)
		{
			//subRL自己就是新增点
			parent->_bf = subR->_bf = 0;
		}
		else if (bf == -1)
		{
			//subRL的左子树上新增
			parent->_bf = 0;
			subRL->_bf = 0;
			subR->_bf = 1;
		}
		else if (bf == 1)
		{
			//subRL的右子树上新增
			parent->_bf = -1;
			subRL->_bf = 0;
			subR->_bf = 0;
		}
		else
		{
			assert(false);
		}
	}
	void RotateLR(Node* parent)      //先左单旋，再右单旋（LR型）
	{
		Node* subL = parent->_left;
		Node* subLR = subL->_right;
		int bf = subLR->_bf;

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

		if (bf == 0)
		{
			//subLR自己就是新增点
			parent->_bf = subL->_bf = 0;
		}
		else if (bf == -1)
		{
			//subLR的左子树上新增
			parent->_bf = 1;
			subLR->_bf = 0;
			subL->_bf = 0;
		}
		else if (bf == 1)
		{
			//subLR的右子树上新增
			parent->_bf = 0;
			subLR->_bf = 0;
			subL->_bf = -1;
		}
		else
		{
			assert(false);
		}
	}

	//中序打印
	void Inorder()
	{
		_Inorder(_root);
		cout << endl;
	}
	

	//检查是否为AVL树
	bool IsBalance()
	{
		return _IsBalance(_root);
	}
	
private:
	void _Inorder(Node* root)
	{
		if (root == nullptr)
			return;
		_Inorder(root->_left);
		cout << root->_kv.first << " ";
		_Inorder(root->_right);
	}
	int _High(Node* root)
	{
		if (root == nullptr)
			return 0;

		int LeftHigh = _High(root->_left);
		int RightHigh = _High(root->_right);
		return LeftHigh > RightHigh ? LeftHigh + 1 : RightHigh + 1;
	}
	bool _IsBalance(Node* root)
	{
		if (root == nullptr)
			return true;
		int LeftHigh = _High(root->_left);
		int RightHigh = _High(root->_right);
		if (RightHigh - LeftHigh != root->_bf)
		{
			cout << _root->_kv.first << "当前节点平衡因子有问题" << endl;
			return false;
		}
		return abs(LeftHigh - RightHigh) < 2
			&& _IsBalance(root->_left)
			&& _IsBalance(root->_right);
	}
	Node* _root = nullptr;
};

源文件

#define   _CRT_SECURE_NO_WARNINGS 1
#include "AVLTree.h"
int main()
{
	int a[] = { 16,3,7,11,9,26,18,14,15 };
	AVLTree<int, int> t;
	for (auto e : a)
	{
		t.Insert(make_pair(e, e));
	}
	t.Inorder();
	cout << "输出1代表是AVL树，输出0代表不是：" << t.IsBalance() << endl;
	
	return 0;
}