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AVL树又称高度平衡的二叉搜索树。
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(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 )
当插入一个节点时，如果这个节点的父节点的平衡因子不满足AVL树的特点，这时就需要对AVL树进行调整，即旋转。
1、左单旋转
2、右单旋转
3、左右旋转
4、右左旋转
实现代码如下：
#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;//平衡因子
AVLTreeNode(const K& key,const V& value)
:_bf(0)
,_left(NULL)
,_right(NULL)
,_parent(NULL)
,_key(key)
,_value(value)
{ }
};
template<class K,class V>
class AVLTree
{
public:
typedef AVLTreeNode<K,V> Node;
AVLTree()
:_root(NULL)
{ }
bool Insert(const K& key,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->_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;
}
//检查树是否平衡
//更新平衡因子，不满足条件时，进行旋转
while (parent)
{
if(cur == parent->_left)
parent->_bf--;
else
parent->_bf++;
if(parent->_bf == 0)
break;
else if(parent->_bf == -1 || parent->_bf == 1)//回朔
{
cur = parent;
parent = cur->_parent;
}
else//平衡因子为2或-2
{
if(cur->_bf == 1)
{
if(parent->_bf == 2)
RotateL(parent);
else//-2
RotateLR(parent);
}
else
{
if(parent->_bf == -2)
RotateR(parent);
else
RotateRL(parent);
}
break;
}
}
return true;
}
Node* Find(const K& key)
{
if(_root == NULL)
return NULL;
Node* cur = _root;
while (cur)
{
if(cur->_key < key)
{
cur = cur->_right;
}
else if(cur->_key > key)
{
cur = cur->_left;
}
else
{
return cur;
}
}
return NULL;
}
void InOrder()
{
_InOrder(_root);
cout<<endl;
}
bool IsBlance()
{
return _IsBlance(_root);
}
~AVLTree()
{
_Destroy(_root);
_root = NULL;
}
protected:
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;
subR->_parent = ppNode;
if(ppNode == NULL)
{
_root = subR;
}
else
{
if(ppNode->_left == parent)
ppNode->_left = subR;
else
ppNode->_right = subR;
}
parent->_bf = subR->_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;
subL->_parent = ppNode;
if(ppNode == NULL)
{
_root = subL;
}
else
{
if(ppNode->_left == parent)
ppNode->_left = subL;
else
ppNode->_right = subL;
}
parent->_bf = subL->_bf = 0;
}
void RotateLR(Node* parent)
{
Node* subL = parent->_left;
Node* subLR =subL->_right;
int bf = subLR->_bf;
RotateL(parent->_left);
RotateR(parent);
//根据subL的平衡因子修正其他节点的平衡因子
if(bf == -1)
{
subL->_bf = 0;
parent->_bf = 1;
}
else if(bf == 1)
{
subL->_bf = -1;
parent->_bf = 0;
}
else
{
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);
//根据subRL的平衡因子修正其它节点的平衡因子
if(bf == 1)
{
subR->_bf = 0;
parent->_bf = -1;
}
else if(bf == -1)
{
subR->_bf = 1;
parent->_bf = 0;
}
else
{
parent->_bf = subR->_bf = 0;
}
}
void _Destroy(Node* root)
{
if(root == NULL)
return;
_Destroy(root->_left);
_Destroy(root->_right);
delete root;
}
void _InOrder(Node* root)
{
if(root == NULL)
return;
_InOrder(root->_left);
cout<<root->_key<<" ";
_InOrder(root->_right);
}
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(Node* root)
{
if(root == NULL)
return true;
int left = _Height(root->_left);
int right = _Height(root->_right);
if((right - left) != root->_bf || abs(right - left) > 1)
{
cout<<"该节点的平衡因子异常:";
cout<<root->_key<<" "<<endl;
return false;
}
return _IsBlance(root->_left) && _IsBlance(root->_right);
}
protected:
Node* _root;
};
void AVLTreeTest()
{
int a[] = {16, 3, 7, 11, 9, 26, 18, 14, 15};
AVLTree<int,int> tree;
for(int i = 0;i < 9;i++)
{
tree.Insert(a[i],i);
}
tree.InOrder();
tree.Find(16);
tree.IsBlance();
}
void TestTree_SP()
{
int a[] = {4, 2, 6, 1, 3, 5, 15, 7, 16, 14};
AVLTree<int, int> t;
for (int i = 0; i < sizeof(a)/sizeof(a[0]); ++i)
{
t.Insert(a[i], i);
}
t.InOrder();
t.IsBlance();
}