- AVL树
AVL树又称为高度平衡的二叉搜索树，是1962年有俄罗斯的数学家G.M.Adel’son-Vel’skii和E.M.Landis提出来的。它能保持二叉树的高度平衡，尽量降低二叉树的高度，减少树的平均搜索长度。
- AVL树的性质
- 左子树和右子树的高度之差的绝对值不超过1
- 树中的每个左子树和右子树都是AVL树
- 每个节点都有一个平衡因子（balance factor bf）,任一节点的平衡因子是1,0，-1.（每个节点的平衡因子等于右子树的高度减去左子树的高度）
- AVL树的效率
一棵AVL树有N个节点，其高度可以保持在log2N，插入/删除/查找的时间复杂度也是log2N。 （ps：log2N是表示log以2为底N的对数，evernote不支持公式。）
//AVLTree.h
#pragma once
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)
{}
};
template<class K,class V>
class AVLTree
{
typedef AVLTreeNode<K,V> Node;
public:
AVLTree()
:_root(NULL)
{}
bool Insert(const K& key,const V& value)
{
//插入节点
if(_root == NULL)
{
_root = new Node(key,value);
}
Node* cur = _root;
Node* parent = NULL;
while(cur)
{
if(cur->_key < key)
{
parent = cur;
cur = cur->_right;
}
else if(cur->_key > key)
{
parent = cur;
cur = cur->_left;
}
else
return false;
}
cur = new Node(key,value);
if(parent->_key < key)
{
parent->_right = cur;
cur->_parent = parent;
}
else
{
parent->_left = cur;
cur->_parent = parent;
}
//更新平衡因子
while(parent)
{
//右孩子平衡因子加1，左孩子平衡因子减1
if(parent->_right == cur)
parent->_bf ++;
else if(parent->_left == cur)
parent->_bf --;
if(parent->_bf == 0)
break;
else if(parent->_bf == 1 || parent->_bf == -1)
{
cur = parent;
parent = cur->_parent;
}
else //_bf = 2 || -2
{
if(parent->_bf == 2) //parent->_bf == 2
{
if(cur->_bf == 1)
RotateL(parent);
else //_bf = -1
RotateRL(parent);
}
else if(parent->_bf == -2) //parent->_bf == -2
{
if(cur->_bf == -1)
RotateR(parent);
else //_bf = 1
RotateLR(parent);
}
return true;
}
}
return true;
}
void RotateL(Node* parent)
{
Node* subR = parent->_right;
Node* subRL = subR->_left;
parent->_right = subRL;
if(subRL)
{
subRL->_parent = parent;
}
subR->_left = parent;
Node* ppNode = parent->_parent;
parent->_parent = subR;
if(ppNode == NULL)
{
_root = subR;
subR->_parent = NULL;
}
else
{
if(ppNode->_left == parent)
{
ppNode->_left = subR;
subR->_parent = ppNode;
}
else
{
ppNode->_right = subR;
subR->_parent = ppNode;
}
}
subR->_bf = parent->_bf = 0;
}
void RotateR(Node* parent)
{
Node* subL = parent->_left;
Node* subLR = subL->_right;
parent->_left = subLR;
if(subLR)
{
subLR->_parent = parent;
}
subL->_right = parent;
Node* ppNode = parent->_parent;
parent->_parent = subL;
//判断parent是否有父亲节点
if(ppNode == NULL)
{
_root = subL;
subL->_parent = NULL;
}
else
{
if(ppNode->_left == parent)
{
ppNode->_left = subL;
subL->_parent = ppNode;
}
else
{
ppNode->_right = subL;
subL->_parent = ppNode;
}
}
subL->_bf = parent->_bf = 0;
}
void RotateRL(Node* parent)
{
Node* subR = parent->_right;
Node* subRL = subR->_left;
int bf = subRL->_bf;
RotateR(parent->_right);
RotateL(parent);
if(bf == 1)
{
subR->_bf = 0;
parent->_bf = -1;
}
else if(bf == -1)
{
parent->_bf = 0;
subR->_bf = 1;
}
else
{
subR->_bf = parent->_bf = 0;
}
subRL->_bf = 0;
}
void RotateLR(Node* parent)
{
Node* subL = parent->_left;
Node* subLR = subL->_right;
int bf = subLR->_bf;
RotateL(parent->_left);
RotateR(parent);
if(bf == 1)
{
subL->_bf = -1;
parent->_bf = 0;
}
else if(bf == -1)
{
parent->_bf = 1;
subL->_bf = 0;
}
else
{
subL->_bf = parent->_bf = 0;
}
subLR->_bf = 0;
}
void InOrder()
{
_InOrder(_root);
cout<<endl;
}
int Height(Node* root)
{
if(root == NULL)
{
return 0;
}
int left = Height(root->_left);
int right = Height(root->_right);
return left > right ? (left+1) : (right+1);
}
bool IsBlance()
{
return _IsBlance(_root);
}
bool _IsBlance(Node* root)
{
if(root == NULL)
{
return true;
}
int left = Height(root->_left);
int right = Height(root->_right);
if((right-left) != root->_bf)
{
cout<<"平衡因子异常"<<root->_key<<endl;
return false;
}
return abs(right-left < 2)
&& _IsBlance(root->_left)
&& _IsBlance(root->_right);
}
void _InOrder(Node* root)
{
if(root == NULL)
{
return;
}
_InOrder(root->_left);
cout<<root->_key<<" ";
_InOrder(root->_right);
}
protected:
Node* _root;
};
void TestTree()
{
AVLTree<int,int> tree;
int a[]={16, 3, 7, 11, 9, 26, 18, 14, 15};
for(size_t i = 0; i < sizeof(a)/sizeof(a[0]); ++i)
{
tree.Insert(a[i],i);
}
tree.InOrder();
cout<<"Blance?"<<tree.IsBlance()<<endl;
getchar();
}
void TestTree1()
{
AVLTree<int,int> tree1;
int array1[]={4, 2, 6, 1, 3, 5, 15, 7, 16, 14};
for(size_t i = 0; i < sizeof(array1)/sizeof(array1[0]); ++i)
{
tree1.Insert(array1[i],i);
}
tree1.InOrder();
cout<<"Blance?"<<tree1.IsBlance()<<endl;
getchar();
}//Test.cpp
#include<iostream>
using namespace std;
#include"AVLTree.h"
int main()
{
//TestTree();
TestTree1();
getchar();
return 0;
}