AVL树浅谈

前言

大家好，我是jiantaoyab，本篇文章给大家介绍AVL树。

基本概念

AVL树（Adelson-Velsky和Landis树）是一种自平衡的二叉搜索树，得名于其发明者G. M. Adelson-Velsky和E. M. Landis。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树特点

二叉搜索树性质：AVL树本质上是一棵二叉搜索树，即每个节点的左子树中的所有节点的值都小于该节点的值，而右子树中的所有节点的值都大于该节点的值。
平衡条件：AVL树的每个节点的左子树和右子树的高度差之绝对值不超过1。这是AVL树与其他二叉搜索树的主要区别，保证了树的平衡性，从而优化了查找、插入和删除操作的性能。
平衡因子：每个节点都有一个平衡因子，定义为该节点的右子树高度减去左子树高度。在AVL树中，平衡因子的取值只能是-1、0或1。

模拟实现

AVL树节点定义

template<class K,class V>
struct AVLTreeNode
{AVLTreeNode<K, V> *_left;AVLTreeNode<K, V> *_right;AVLTreeNode<K, V> *_parent;pair<K,V> _kv;  //pair是将2个数据组合成一组数据int _bf; //balance factorAVLTreeNode(const pair<K, V>&kv):_left(nullptr), _right(nullptr), _parent(nullptr), _bf(0), _kv(kv){}};

插入操作

bool Insert(const pair<K, V>&kv){if (_root == nullptr){_root = new Node (kv);return true;}//二叉搜索树插入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;}//控制平衡//1、更新平衡因子 //2、异常，旋转平衡处理//只会影响这条路径，最坏更新到根while (parent){if (cur == parent->_left){parent->_bf--;}else{parent->_bf++;}if (parent->_bf == 0){break;}//继续更新else if (parent->_bf == 1 || parent->_bf == -1){cur = parent;parent = parent->_parent;}//旋转处理 //1：树平衡了//2:树的整体高度降了1else if (parent->_bf == 2 || parent->_bf == -2){//右单旋if (parent->_bf == -2 && cur->_bf == -1){RotateR(parent);}//左单旋else if (parent->_bf == 2 && cur->_bf == 1){RotateL(parent);}//双旋左右else if (parent->_bf == -2 && cur->_bf == 1){RotateLR(parent);}//双旋右左else if (parent->_bf == 2 && cur->_bf == -1){RotateRL(parent);}else{assert(false);}//旋转完之后不影响上面的，可以直接退出break;	}//插入之前，平衡因子有问题else{assert(false);}}return true;
}

旋转操作

右单旋

	//右单旋void RotateR(Node* parent){Node* subL = parent->_left;Node* subLR = subL->_right;parent->_left = subLR;subL->_right = parent;//更新父节点if (subLR)subLR->_parent = parent;Node* parentParent = parent->_parent;parent->_parent = subL;//如果原来是根if (parent == _root){_root = subL;_root->_parent = nullptr;}//如果是别人的子树else{if (parentParent->_left == parent)parentParent->_left = subL;elseparentParent->_right = subL;subL->_parent = parentParent;}subL->_bf = parent->_bf = 0;}

左单旋

void RotateL(Node* parent){Node* subR = parent->_right;Node* subRL = subR->_left;parent->_right = subRL;if (subRL){subRL->_parent = parent;}Node* parentParent = parent->_parent;subR->_left = parent;parent->_parent = subR;//原来是根if (_root == parent){_root = subR;subR->_parent = nullptr;}//原来是别人的子树else{if (parentParent->_left == parent)parentParent->_left = subR;elseparentParent->_right = subR;subR->_parent = parentParent;}subR->_bf = 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){parent->_bf = 1;subL->_bf = 0;}else if (bf == 1){parent->_bf = 0;subL->_bf = -1;}else{parent->_bf = subL->_bf = 0;}subLR->_bf = 0;}

右左双旋

void RotateRL(Node *parent)	{//   30 (parent)//  /   \//  a   90(subR)//      /   \     //    60(subRL)   d//   b  cNode *subR = parent->_right;Node *subRL = subR->_left;int bf = subRL->_bf;RotateR(parent->_right);RotateL(parent);//这里看图 变化后的图if (bf == 1){parent->_bf = -1;subR->_bf = 0;subRL->_bf = 0;}else if (bf == -1){parent->_bf = 0;subR->_bf = 1; subRL->_bf = 0;}else if (bf == 0){parent->_bf = 0;subR->_bf = 0;subRL->_bf = 0;}else{assert(false);}}

判断是不是AVL树

int Height(Node *root){if (root == NULL)return 0;int leftHeight = Height(root->_left);int rightHeight = Height(root->_right);return leftHeight > rightHeight ? leftHeight + 1 : rightHeight + 1;}bool IsBalance(){return _IsBalance(_root);}//1.检查是不是搜索二叉树//2.检查每一课子树是不是AVL树bool _IsBalance(Node *root){if (root == NULL)return true;//当前树检查int leftHeight = Height(root->_left);int rightHeight = Height(root->_right);//平衡因子出问题if (rightHeight - leftHeight != root->_bf){cout << root->_kv.first << "NOW::" << root->_bf << endl;cout << root->_kv.first << "CORRECT::" << rightHeight - leftHeight << endl;return false;}//左右高度差不能超过2return abs(rightHeight - leftHeight) < 2 && _IsBalance(root->_left)&& _IsBalance(root->_right);}