目录

AVL树的插入(包含更新平衡因子)

子树的旋转

子树的左旋转

子树的右旋转

左右双旋

 右左双旋

求树的高度

判断是否为平衡二叉树

层序遍历

代码总览

AVL树的插入(包含更新平衡因子)

bool Insert(const pair& kv)
	{
		//1.搜索树的规则插入
		//2.看是是否违反平衡规则，如果违反就需要处理：旋转
		if (_root == nullptr)
		{
			_root = new Node(kv);
			_root->_bf = 0;
			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;
		}
		else
		{
			parent->_left = cur;
		}
		cur->_parent = parent;

		//...
		//更新平衡因子
		while (parent)//最远要更新根
		{
			if (cur == parent->_right)
			{
				parent->_bf++;
			}
			else
			{
				parent->_bf--;
			}
			
			//是否继续更新？
			if (parent->_bf == 0)//1 or -1 ->插入节点填上了矮的那边
			{
				//高度不变，更新结束
				break;
			}
			else if(parent->_bf == 1 || parent->_bf == -1)//0 -> 1 or -1插入节点导致一边变高了 
			{
				//子树的高度变了，继续更新祖先
				cur = cur->_parent;
				parent = parent->_parent;
			}
			else if (parent->_bf == 2 || parent->_bf == -2)
			//-1 or 1 -> 2or-2 插入节点导致一边变高了
			{
				//子树不平衡 -- 需要旋转处理
				//...
				if (parent->_bf ==  2 && cur->_bf == 1)//左单旋
				{
					RotateL(parent);
				}
				else if (parent->_bf == 2 && cur->_bf == 1)//右单选
				{
					RotateR(parent);
				}
				else if (parent->_bf == -2 && cur->_bf == -1)//左右双旋
				{
					RotateLR(parent);
				}
				else if (parent->_bf == 2 && cur->_bf == -1)//右左双旋
				{
					RotateRL(parent);
				}

				break;
			}
			else
			{
				//插入之前AVL树就存在不平衡子树，平衡因子的绝对值>=2的节点
				assert(false);
			}
		}

		return true;
	}

子树的旋转

子树的左旋转

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

		parent->_right = subRL;

		if(subRL)
			subRL->_parent = parent;//易错

		Node* ppNode = parent->_parent;

		subR->_left = parent;

		parent->_parent = subR;//易错

		if (parent == _root)
		{
			_root = subR;
			_root->_parent = nullptr;
		}
		else
		{
			if (parent == ppNode->_left)
			{
				ppNode->_left = subR;
			}
			else
			{
				ppNode->_right = subR;
			}
			subR->_parent = ppNode;
		}
		//更新平衡因子
		parent->_bf = 0;
		subR->_bf = 0;
	}

 

子树的右旋转

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

		parent->_left = subLR;

		if (subLR)
			subLR->_parent = parent;//易错

		Node* ppNode = parent->_parent;

		subL->_right = parent;

		parent->_parent = subL;//易错

		if (parent == _root)
		{
			_root = subL;
			_root->_parent = nullptr;
		}
		else
		{
			if (parent == ppNode->_left)
			{
				ppNode->_left = subL;
			}
			else
			{
				ppNode->_right = subL;
			}
			subL->_parent = ppNode;
		}
		//更新平衡因子
		parent->_bf = 0;
		subL->_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);

		//更新平衡因子
		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
		{
			//subLR->_bf旋转前就有问题
			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)
		{
			subRL->_bf = 0;
			parent->_bf = 0;
			subR->_bf = 0;
		}
		else if (bf == 1)
		{
			subRL->_bf = 0;
			parent->_bf = -1;
			subR->_bf = 0;
		}
		else if (bf == -1)
		{
			subRL->_bf = 0;
			parent->_bf = 0;
			subRL->_bf = 1;
		}
		else
		{
			//subLR->_bf旋转前就有问题
			assert(false);
		}
	}

求树的高度

	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;
	}


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

判断是否为平衡二叉树

	bool _IsBalanceTree(Node* root)
	{
		// 空树也是AVL树
		if (nullptr == root)
			return true;

		// 计算pRoot节点的平衡因子：即pRoot左右子树的高度差
		int leftHeight = _Height(root->_left);
		int rightHeight = _Height(root->_right);
		int diff = rightHeight - leftHeight;

		// 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 (abs(diff) >= 2)
		{
			cout << root->_kv.first << "节点平衡因子异常" << endl;
			return false;
		}

		if (diff != root->_bf)
		{
			cout << root->_kv.first << "节点平衡因子不符合实际" << 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);
	}


	bool IsBalanceTree()
	{
		return _IsBalanceTree(_root);
	}

层序遍历

vector> levelOrder() {
		vector> vv;
		if (_root == nullptr)
			return vv;

		queueq;
		int levelSize = 1;
		q.push(_root);

		while (!q.empty())
		{
			// levelSize控制一层一层出
			vector levelV;
			while (levelSize--)
			{
				Node* front = q.front();
				q.pop();
				levelV.push_back(front->_kv.first);
				if (front->_left)
					q.push(front->_left);

				if (front->_right)
					q.push(front->_right);
			}
			vv.push_back(levelV);
			for (auto e : levelV)
			{
				cout << e << " ";
			}
			cout << endl;

			// 上一层出完，下一层就都进队列
			levelSize = q.size();
		}

		return vv;
	}

代码总览

#pragma once
#include
#include
#include
#include

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

	//左右子树高度差
	int _bf; //balence factor 平衡因子

	AVLTreeNode(const pair& kv)//构造函数
		:_kv(kv)
		, _left(nullptr)
		, _right(nullptr)
		, _parent(nullptr)
		,_bf(0)
	{}

	//AVL树并没有规定必须要实现平衡因子
	//只是一个实现的选择，方便控制平衡
};

template
class AVLTree
{
	typedef AVLTreeNodeNode;
public:
	bool Insert(const pair& kv)
	{
		//1.搜索树的规则插入
		//2.看是是否违反平衡规则，如果违反就需要处理：旋转
		if (_root == nullptr)
		{
			_root = new Node(kv);
			_root->_bf = 0;
			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;
		}
		else
		{
			parent->_left = cur;
		}
		cur->_parent = parent;

		//...
		//更新平衡因子
		while (parent)//最远要更新根
		{
			if (cur == parent->_right)
			{
				parent->_bf++;
			}
			else
			{
				parent->_bf--;
			}
			
			//是否继续更新？
			if (parent->_bf == 0)//1 or -1 ->插入节点填上了矮的那边
			{
				//高度不变，更新结束
				break;
			}
			else if(parent->_bf == 1 || parent->_bf == -1)//0 -> 1 or -1插入节点导致一边变高了 
			{
				//子树的高度变了，继续更新祖先
				cur = cur->_parent;
				parent = parent->_parent;
			}
			else if (parent->_bf == 2 || parent->_bf == -2)
			//-1 or 1 -> 2or-2 插入节点导致一边变高了
			{
				//子树不平衡 -- 需要旋转处理
				//...
				if (parent->_bf ==  2 && cur->_bf == 1)//左单旋
				{
					RotateL(parent);
				}
				else if (parent->_bf == 2 && cur->_bf == 1)//右单选
				{
					RotateR(parent);
				}
				else if (parent->_bf == -2 && cur->_bf == -1)//左右双旋
				{
					RotateLR(parent);
				}
				else if (parent->_bf == 2 && cur->_bf == -1)//右左双旋
				{
					RotateRL(parent);
				}

				break;
			}
			else
			{
				//插入之前AVL树就存在不平衡子树，平衡因子的绝对值>=2的节点
				assert(false);
			}
		}

		return true;
	}

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

		parent->_right = subRL;

		if(subRL)
			subRL->_parent = parent;//易错

		Node* ppNode = parent->_parent;

		subR->_left = parent;

		parent->_parent = subR;//易错

		if (parent == _root)
		{
			_root = subR;
			_root->_parent = nullptr;
		}
		else
		{
			if (parent == ppNode->_left)
			{
				ppNode->_left = subR;
			}
			else
			{
				ppNode->_right = subR;
			}
			subR->_parent = ppNode;
		}
		//更新平衡因子
		parent->_bf = 0;
		subR->_bf = 0;
	}

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

		parent->_left = subLR;

		if (subLR)
			subLR->_parent = parent;//易错

		Node* ppNode = parent->_parent;

		subL->_right = parent;

		parent->_parent = subL;//易错

		if (parent == _root)
		{
			_root = subL;
			_root->_parent = nullptr;
		}
		else
		{
			if (parent == ppNode->_left)
			{
				ppNode->_left = subL;
			}
			else
			{
				ppNode->_right = subL;
			}
			subL->_parent = ppNode;
		}
		//更新平衡因子
		parent->_bf = 0;
		subL->_bf = 0;
	}

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

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

		//更新平衡因子
		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
		{
			//subLR->_bf旋转前就有问题
			assert(false);
		}
	}

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

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

		if (bf == 0)
		{
			subRL->_bf = 0;
			parent->_bf = 0;
			subR->_bf = 0;
		}
		else if (bf == 1)
		{
			subRL->_bf = 0;
			parent->_bf = -1;
			subR->_bf = 0;
		}
		else if (bf == -1)
		{
			subRL->_bf = 0;
			parent->_bf = 0;
			subRL->_bf = 1;
		}
		else
		{
			//subLR->_bf旋转前就有问题
			assert(false);
		}
	}


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

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


	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;
	}



	bool _IsBalanceTree(Node* root)
	{
		// 空树也是AVL树
		if (nullptr == root)
			return true;

		// 计算pRoot节点的平衡因子：即pRoot左右子树的高度差
		int leftHeight = _Height(root->_left);
		int rightHeight = _Height(root->_right);
		int diff = rightHeight - leftHeight;

		// 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 (abs(diff) >= 2)
		{
			cout << root->_kv.first << "节点平衡因子异常" << endl;
			return false;
		}

		if (diff != root->_bf)
		{
			cout << root->_kv.first << "节点平衡因子不符合实际" << 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);
	}

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

	vector> levelOrder() {
		vector> vv;
		if (_root == nullptr)
			return vv;

		queueq;
		int levelSize = 1;
		q.push(_root);

		while (!q.empty())
		{
			// levelSize控制一层一层出
			vector levelV;
			while (levelSize--)
			{
				Node* front = q.front();
				q.pop();
				levelV.push_back(front->_kv.first);
				if (front->_left)
					q.push(front->_left);

				if (front->_right)
					q.push(front->_right);
			}
			vv.push_back(levelV);
			for (auto e : levelV)
			{
				cout << e << " ";
			}
			cout << endl;

			// 上一层出完，下一层就都进队列
			levelSize = q.size();
		}

		return vv;
	}

	bool IsBalanceTree()
	{
		return _IsBalanceTree(_root);
	}

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

private:
	Node* _root = nullptr;
};

void TestAVLTree1()
{
	//int a[] = { 1, 2, 3, 4, 5, 6, 7, 8 };
	int a[] = { 30,29,28,27,26,25,24,11,8,7,6,5,4,3,2,1 };
	AVLTree t;
	for (auto e : a)
	{
		t.Insert(make_pair(e, e));
	}
	t.levelOrder();
}

void TestAVLTree2()
{
	const size_t N = 1024 * 1024 * 10;
	vector v;
	v.reserve(N);
	srand(time(0));
	for (size_t i = 0; i < N; ++i)
	{
		//v.push_back(rand());
		v.push_back(i);
	}

	AVLTree t;
	for (auto e : v)
	{
		t.Insert(make_pair(e, e));
	}

	//t.levelOrder();
	//cout << endl;
	cout << "是否平衡?" << t.IsBalanceTree() << endl;
	cout << "高度:" << t.Height() << endl;


	//t.InOrder();
}