【C++：AVL树】深入理解AVL树的平衡之道：从原理、旋转到完整实现代码

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

// AVL树的结构
template<class K,class V>
struct AVLTreeNode
{
	// 需要parent指针，后序更新平衡因子的时候，就可以看到
	pair<K, V> _kv;
	AVLTreeNode<K, V>* _left;
	AVLTreeNode<K, V>* _right;
	AVLTreeNode<K, V>* _parent;
	int _bf; // balance factor：平衡因子

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

template<class K, class V>
struct AVLTree
{
	typedef AVLTreeNode<K, V> Node;
public:
	// 这里AVL树的插入和搜索二叉树的插入规则是一样的，就多了一个平衡因子（右子树高度 - 左子树高度）
	bool Insert(const pair<K, V>& kv)
	{
		if (_root == nullptr)
		{
			_root = new Node(kv); // 这里把搜索二叉树那里的key改成kv（pair的结构，包含key和value）
			return true;
		}

		Node* parent = nullptr;
		Node* cur = _root;
		while (cur)
		{
			// 比kv_first大就往右边走，比kv_first小就往左边走，相等就return false
			if (cur->_kv.first < kv.first) // pair结果做kv时，first是key，second是value
			{
				parent = cur;
				cur = cur->_right;
			}
			else if (cur->_kv.first > kv.first)
			{
				parent = cur;
				cur = cur->_left;
			}
			else
			{
				return false;
			}
		}

		cur = new Node(kv);
		// cur->_bf = 0;
		if (parent->_kv.first < kv.first)
		{
			parent->_right = cur;
		}
		else
		{
			parent->_left = cur;
		}
		// 构建好了三叉链结构：父亲指向孩子，孩子反过来也指向了父亲
		cur->_parent = parent;

		// 控制平衡：对平衡要进行单独的控制
		// 1、更新平衡因子
		while (parent)
		{
			if (cur == parent->_left)
			{
				parent->_bf--;
			}
			else
			{
				parent->_bf++;
			}

			if (parent->_bf == 0)
			{
				// parent所在的子树的高度不变，不会再影响上一层，更新结束
				break;
			}
			else if (parent->_bf == 1 || parent->_bf == -1)
			{
				// parent所在的子树的高度不变，会再影响上一层，继续向上更新
				cur = parent;
				parent = parent->_parent;
			}
			else if (parent->_bf == 2 || parent->_bf == -2)
			{
				// parent所在的子树已经不平衡了，需要旋转处理
				if (parent->_bf == -2 && cur->_bf == -1)
				{
					RotateR(parent);
				}
				else if (parent->_bf == 2 && cur->_bf == 1)
				{
					RotateL(parent);
				}
				else if (parent->_bf == -2 && cur->_bf == 1)
				{
					RotateLR(parent);
				}
				else if (parent->_bf == 2 && cur->_bf == -1)
				{
					RotateRL(parent);
				}
				else
				{
					assert(false);
				}

				break;
			}
			else
			{
				assert(false);
			}
		}

		return true;
	}

	// 右单旋
	void RotateR(Node* parent)
	{
		Node* subL = parent->_left;
		Node* subLR = subL->_right;

		parent->_left = subLR;
		if (subLR)
			subLR->_parent = parent;

		Node* parentParent = parent->_parent;

		subL->_right = parent;
		parent->_parent = subL;

		if (parent == _root) // 根节点
		{
			_root = subL;
			subL->_parent = nullptr;
		}
		else
		{
			if (parentParent->_left == parent) // 如果是子树
			{
				parentParent->_left = subL;
			}
			else
			{
				parentParent->_right = subL;   // 如果是根节点
			}

			subL->_parent = parentParent;
		}

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

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

		//parent->_right = subR->_left;
		parent->_right = subRL;
		if (subRL)
			subRL->_parent = parent;
			//subRL->_right;

		Node* parentParent = parent->_parent;

		subR->_left = parent;
		parent->_parent = subR;

		//if (parentParent == nullptr)
		if (parent == _root)
		{
			_root = subR;
			subR->_parent = nullptr;
		}
		else
		{
			if (parent == parentParent->_left)
			{
				parentParent->_left = subR;
			}
			else
			{
				parentParent->_right = subR;
			}
			subR->_parent = parentParent;
		}
		parent->_bf = subR->_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);

		// 平衡因子的条件
		if (bf == 0)
		{
			parent->_bf = 0;
			subL->_bf = 0;
			subLR->_bf = 0;
		}
		else if (bf == 1)
		{
			parent->_bf = 0;
			subL->_bf = -1;
			subLR->_bf = 0;
		}
		else if (bf == -1)
		{
			parent->_bf = 1;
			subL->_bf = 0;
			subLR->_bf = 0;
		}
		else
		{
			assert(false);
		}
	}

	// 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 == 0)
		{
			subR->_bf = 0;
			subRL->_bf = 0;
			parent->_bf = 0;
		}
		else if (bf == 1)
		{
			subR->_bf = 0;
			subRL->_bf = 0;
			parent->_bf = -1;
		}
		else if (bf == -1)
		{
			subR->_bf = 1;
			subRL->_bf = 0;
			parent->_bf = 0;
		}
		else
		{
			assert(false);
		}
	}

	// 查找
	Node* Find(const K& key)
	{
		Node* cur = _root;
		while (cur)
		{
			if (cur->_kv.first < key)
			{
				cur = cur->_right;
			}
			else if (cur->_kv.first > key)
			{
				cur = cur->_left;
			}
			else
			{
				return cur;
			}
		}
		return nullptr;
	}

	int Size()
	{
		return _Size(_root);
	}

	int Height()
	{
		return _Height(_root);
	}

	bool IsBalanceTree()
	{
		return _IsBalanceTree(_root); // 不方便放根节点，所以套一层娃
	}

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

private:
	// 传根，不能直接传，外部调用内部
	int _Size(Node* root)
	{
		if (root == nullptr)
			return 0;

		return _Size(root->_left) + _Size(root->_right) + 1;
	}

	// 求高度
	int _Height(Node* root)
	{
		if (root == nullptr)
			return 0;
		int lh = _Height(root->_left);
		int rh = _Height(root->_right);
		return lh > rh ? lh + 1 : rh + 1; // 后序求
	}

	// 实现要写成递归，IsBalance
	// 递归计算平衡因子绝对值是否<=2
	bool _IsBalanceTree(Node* root)
	{
		// 空树也是AVL树 
		if (nullptr == root)
			return true;
		// 计算pRoot结点的平衡因子：即pRoot左右子树的高度差 
		// 求出左右子树的高度
		int lh = _Height(root->_left);
		int rh = _Height(root->_right);

		// 如果计算出的平衡因子与pRoot的平衡因子不相等，或者 
		// pRoot平衡因子的绝对值超过1，则一定不是AVL树 
		if (rh - lh != root->_bf || abs(root->_bf) >= 2)
		{
			cout << root->_kv.first << ":平衡因子异常" << endl;
			return false;
		}
		// pRoot的左和右如果都是AVL树，则该树⼀定是AVL树 
		return _IsBalanceTree(root->_left) && _IsBalanceTree(root->_right); // 递归检测左子树和右子树
	}

	int _InOrder(Node* root)
	{
		if (root == nullptr)
			return 0;

		_InOrder(root->_left);
		cout << root->_kv.first << ":" << root->_kv.second << endl;
		_InOrder(root->_right);
	}

private:
	Node* _root = nullptr;
};

#define  _CRT_SECURE_NO_WARNINGS  1

#include"AVLTree.h"

void TestAVLTree1()
{
	AVLTree<int, int> t;
	// 常规的测试例 
	//int a[] = { 16, 3, 7, 11, 9, 26, 18, 14, 15 };
	// 特殊的带有双旋场景的测试用例 
	int a[] = { 4, 2, 6, 1, 3, 5, 15, 7, 16, 14 };

	for (auto e : a)
	{
		if (e == 14)
		{
			int i = 0;
		}
		t.Insert({ e,e });
		cout << e << "->" << t.IsBalanceTree() << endl;
	}
	//t.InOrder();

	cout << t.IsBalanceTree() << endl;
}

// 插入一堆随机值，测试平衡，顺便测试一下高度和性能
void TestAVLTree2()
{
	const int N = 100000;
	vector<int> v;
	v.reserve(N);
	srand(time(0));
	for (size_t i = 0; i < N; i++)
	{
		v.push_back(rand() + i);
	}
	size_t begin2 = clock();
	AVLTree<int, int> t;
	for (auto e : v)
	{
		t.Insert(make_pair(e, e));
	}
	size_t end2 = clock();
	cout << "Insert:" << end2 - begin2 << endl;
	cout << t.IsBalanceTree() << endl;

	cout << "Height:" << t.Height() << endl;
	cout << "Size:" << t.Size() << endl;
	size_t begin1 = clock();
	// 确定在的值 
	/*for (auto e : v)
	{
	t.Find(e);
	}*/
	// 随机值 
	for (size_t i = 0; i < N; i++)
	{
		t.Find((rand() + i));
	}
	size_t end1 = clock();
	cout << "Find:" << end1 - begin1 << endl;
}

int main()
{
	TestAVLTree1();
	//TestAVLTree2();

	return 0;
}