AVL树又称高度平衡的二叉搜索树，是1962年俄罗斯的数学家提出来的。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的性质：
（1）左子树和右子树的高度之差的绝对值不超过1。
（2）树中的每个左子树和右子树都是AVL树。
（3）每个节点都有一个平衡因子，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）。
代码实现如下：
#include<iostream>
using namespace std;
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; //平衡因子
AVLTreeNode(const K& key,const V& value)
:_key(key)
, _value(value)
, _left(NULL)
, _right(NULL)
, _parent(NULL)
, _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)
{
if (_root == NULL)
{
_root = new Node(key,value);
return true;
}
Node* cur = _root;
Node* parent = NULL;
while (cur)
{
if (cur->_key > key)
{
parent = cur;
cur = cur->_left;
}
else if (cur->_key < key)
{
parent = cur;
cur = cur->_right;
}
else
{
return false;
}
}
cur = new Node(key,value);
if (parent->_key > key)
{
parent->_left = cur;
cur->_parent = parent;
}
else
{
parent->_right = cur;
cur->_parent = parent;
}
//更新平衡因子
//不平衡，则进行旋转
while (parent)
{
if (parent->_right==cur)
parent->_bf++;
else
parent->_bf--;
//父节点平衡因子为0时，退出（说明父节点的两边高度一样，算路径长度的话都一样，没有影响）
if (parent->_bf == 0)
break;
//父节点平衡因子为1或-1的时候（说明是从0+1或0-1得来的），父节点两边高度不同，故需要继续更新平衡因子
else if (parent->_bf == 1 || parent->_bf == -1)
{
cur = parent;
parent = cur->_parent;
}
//父节点平衡因子为2或-2时，旋转
else //(parent->_bf==2||parent->_bf==-2) 旋转
{
if (parent->_bf == -2)
{
if (cur->_bf == -1)//右单旋
{
_RotateR(parent);
}
else //(cur->_bf==1) 左右单旋
{
_RotateLR(parent);
}
}
else //(parent->_bf==2)
{
if (cur->_bf == 1) //左单旋
{
_RotateL(parent);
}
else //(cur->_bf==-1)右左单旋
{
_RotateRL(parent);
}
}
break;
}
}
}
Node* Find(const K& key)
{
if (_root == NULL)
return false;
Node* cur = _root;
while (cur)
{
if (cur->_key > key)
cur = cur->_left;
else if (cur->_key < key)
cur = cur->_right;
else
return cur;
}
return false;
}
bool Remove(const K& key)
{
if (_root == NULL)
return false;
Node* parent = NULL;
Node* cur = _root;
while (cur)
{
if (cur->_key > key)
{
parent = cur;
cur = cur->_left;
}
else if (cur->_key < key)
{
parent = cur;
cur = cur->_right;
}
else
{
Node* del;
if (cur->_right == NULL)
{
del = cur;
if (parent == NULL)
{
_root = cur->_left;
//_root->_bf = 0;
}
else
{
if (parent->_left == cur)
{
parent->_left = cur->_left;
parent->_bf++;
}
else
{
parent->_right = cur->_left;
parent->_bf--;
}
}
delete del;
}
else if (cur->_left == NULL)
{
del = cur;
if (parent == NULL)
{
_root = cur->_right;
_root->_bf = 0;
}
else
{
if (parent->_left == cur)
{
parent->_left = cur->_right;
parent->_bf++;
}
else
{
parent->_right = cur->_right;
parent->_bf--;
}
}
delete del;
}
else
{
parent = cur;
Node* left = cur->_right;
while (left->_left)
{
parent = left;
left = left->_left;
}
del = left;
cur->_key = left->_key;
cur->_value = left->_value;
if (parent->_left == left)
{
parent->_left = left->_right;
parent->_bf++;
}
else
{
parent->_right = left->_right;
parent->_bf--;
}
delete del;
}
break;
}
}
if (cur == NULL)
{
return false;
}
while (parent)
{
if (parent->_bf == 0)
{
break;
}
else if (parent->_bf == 1 || parent->_bf == -1)
{
break;
}
else //parent->_bf=2||parent->_bf=-2
{
if (parent->_bf == -2)
{
if (cur->_bf == -1)
_RotateR(parent);
else //cur->_bf=1
_RotateLR(parent);
}
else
{
if (cur->_bf == 1)
_RotateL(parent);
else
_RotateRL(parent);
}
break;
}
}
return true;
}
void InOrder()
{
_InOrder(_root);
cout << endl;
}
//判断这棵树是否是平衡搜索树
bool IsBlance()
{
return _IsBlance(_root);
}
protected:
void _RotateR(Node* parent)
{
Node* subL = parent->_left;
Node* subLR=subL->_right;
parent->_left = subLR;
if (subLR != NULL)
{
subLR->_parent = parent;
}
subL->_right = parent;
Node* ppNode = parent->_parent;
parent->_parent = subL;
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 _RotateL(Node* parent)
{
Node* subR = parent->_right;
Node* subRL= subR->_left;
parent->_right = subRL;
if (subRL != NULL)
{
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 _RotateRL(Node* parent)
{
Node* subR = parent->_right;
Node* subRL= subR->_left;
int bf = subRL->_bf;
_RotateR(parent->_right);
_RotateL(parent);
if (bf == 1) //从subRL的右边插入
{
parent->_bf = -1;
subR->_bf = 0;
}
else if (bf == -1) //从subRL的左边插入
{
parent->_bf = 0;
subR->_bf = 1;
}
else //(bf=0)
{
parent->_bf = 0;
subR->_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)
{
parent->_bf = 0;
subL->_bf = -1;
}
else if (bf == -1)
{
parent->_bf = 1;
subL->_bf = 0;
}
else //bf=0
{
parent->_bf = 0;
subL->_bf = 0;
}
subLR->_bf = 0;
}
bool _IsBlance(Node* root)
{
if (root == NULL)
return true;
int right = _Height(root->_right);
int left = _Height(root->_left);
if (right - left != root->_bf || abs(right - left) >= 2)
{
cout << "平衡因子异常" << root->_key << endl;
}
return _IsBlance(root->_left) && _IsBlance(root->_right);
}
int _Height(Node* root)
{
if (root == NULL)
return 0;
int right = _Height(root->_right);
int left = _Height(root->_left);
if (right > left)
return (right + 1);
else
return (left + 1);
}
void _InOrder(Node* root)
{
if (root == NULL)
{
return;
}
else
{
_InOrder(root->_left);
cout << root->_key << " ";
_InOrder(root->_right);
}
}
protected:
Node* _root;
};
#include "AVLTree.h"
void Test1()
{
int a[9] = { 16, 3, 7, 11, 9, 26, 18, 14, 15 };
AVLTree<int, int> avl;
for (int i = 0; i < sizeof(a) / sizeof(a[0]); ++i)
{
avl.Insert(a[i],i);
}
avl.InOrder();
cout<<avl.IsBlance()<<endl;
AVLTreeNode<int, int>* ret1 = avl.Find(18);
if (ret1)
cout << ret1->_key << ":" << ret1->_value << endl;
else
cout << "不存在ret1" << endl;
AVLTreeNode<int, int>* ret2 = avl.Find(1);
if (ret2)
cout << ret2->_key << ":" << ret2->_value << endl;
else
cout << "不存在ret2" << endl;
avl.Remove(26);
avl.Remove(18);
avl.Remove(15);
avl.InOrder();
avl.Remove(3);
cout << avl.Remove(7) << endl;
avl.Remove(7);
avl.Remove(9);
avl.Remove(11);
avl.Remove(14);
avl.Remove(15);
cout << avl.Remove(100) << endl;
avl.Remove(16);
avl.Remove(18);
avl.Remove(26);
avl.InOrder();
}
void Test2()
{
int a[10] = { 4, 2, 6, 1, 3, 5, 15, 7, 16, 14 };
AVLTree<int, int> avl;
for (int i = 0; i < sizeof(a) / sizeof(a[0]); ++i)
{
avl.Insert(a[i], i);
}
avl.InOrder();
cout << avl.IsBlance() << endl;
AVLTreeNode<int, int>* ret1 = avl.Find(5);
if (ret1)
cout << ret1->_key << ":" << ret1->_value << endl;
else
cout << "不存在ret1" << endl;
AVLTreeNode<int, int>* ret2 = avl.Find(88);
if (ret2)
cout << ret2->_key << ":" << ret2->_value << endl;
else
cout << "不存在ret2" << endl;
avl.Remove(14);
avl.Remove(16);
avl.Remove(7);
avl.InOrder();
avl.Remove(15);
avl.Remove(6);
avl.Remove(5);
cout << avl.Remove(4) << endl;
avl.Remove(4);
avl.Remove(3);
avl.Remove(2);
avl.Remove(1);
cout << avl.Remove(100) << endl;
avl.Remove(7);
avl.Remove(16);
avl.InOrder();
}
int main()
{
Test1();
cout << endl;
cout << endl;
Test2();
return 0;
}实现结果：
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