【数据结构】中的平衡搜索树-AVLTree
数据结构中：
AVL树是最先发明的自平衡二叉查找树。在AVL树中任何节点的两个子树的高度最大差别为一，所以它也被称为高度平衡树。
特点：
AVL树本质上还是一棵二叉搜索树，它的特点是：
1.本身首先是一棵二叉搜索树。
2.带有平衡条件：每个结点的左右子树的高度之差的绝对值（平衡因子）最多为1。
也就是说，AVL树，本质上是带了平衡功能的二叉查找树（二叉排序树，二叉搜索树）。
在AVL树里面，有几个重要的代码实现：
插入：
bool Insert(const K& key, const V& value)
{
if (_root == NULL)
{
_root = new Node(key, value);
return true;
}
Node* parent = NULL;
Node* cur = _root;
//判断插入的值和cur的大小
while (cur)
{
parent = cur;
if (cur->_key < key)
{
cur = cur->_right;
}
else if (cur->_key>key)
{
cur = cur->_left;
}
else
{
return false;
}
}
cur = new Node(key, value);//构建一个新的节点
if (parent->_key > key)//如果父节点的值大于key,则key就为父节点的左边
{
parent->_left = cur;
cur->_parent = parent;
}
else // 如果父节点的值小于key, 则key就为父节点的右边
{
parent->_right = cur;
cur->_parent = parent;
}
while (parent)
{
if (cur == parent->_left)//如果插入的节点在parent的左边，则平衡因子-1
{
parent->_bf--;
}
else//如果插入的节点在parent的右边，则平衡因子+1
{
parent->_bf++;
}
if (parent->_bf == 0)//平衡因子为0，子树的高度没有变化
{
break;
}
else if(parent->_bf == -1 || parent->_bf == 1)
{
//子树的高度变化，但是高度没有变化，继续向上一层树进行检查，是否对上一层有影响
cur = parent;
parent = cur->_parent;
}
else//子树的高度为2或者-2 导致二叉树不再平衡，则需要进行旋转
{
if(parent->_bf==2)//说明右边子树的高度增加了
{
Node* subR = parent->_right;
if (subR->_bf == 1)
{
RotateL(parent);
}
else
{
RotateRL(parent);
}
}
else//说明左边的子树的高度增加了
{
Node* subL = parent->_left;
if (subL->_bf == -1)
{
RotateR(parent);
}
else
{
RotateLR(parent);
}
}
break;
}
}
}
左单旋：
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)
{
subR = _root;
subR->_parent = NULL;
}
else
{
if (ppNode->_left == parent)
{
ppNode->_left = subR;
}
else
{
ppNode->_right = subR;
}
subR->_parent = ppNode;
}
parent->_bf = subR->_bf = 0;
}
右单旋：
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;
if (ppNode == NULL)
{
_root = subL;
subL->_parent = NULL;
}
else
{
if (ppNode->_left == parent)
{
ppNode->_left = subL;
}
else
{
ppNode->_right = subL;
}
subL->_parent = ppNode;
}
parent->_bf = subL->_bf = 0;
}
左右双旋：
void RotateLR(Node* parent)
{
Node* subL = parent->_left;
Node* subLR = subL->_right;
int bf = subLR->_bf;
RotateL(subL);
RotateR(parent);
if (bf == 0)
{
parent->_bf = subL->_bf = subLR->_bf = 0;
}
else if (bf==1)
{
subL->_bf = -1;
parent->_bf = 0;
subLR->_bf = 0;
}
else
{
subL->_bf = 0;
subLR->_bf = 0;
parent->_bf = 1;
}
}
右左双旋：
void RotateRL(Node* parent)
{
Node* subR = parent->_right;
Node* subRL = subR->_left;
int bf = subRL->_bf;
RotateR(subR);
RotateL(parent);
if (bf == 0)
{
parent->_bf = subR->_bf = subRL->_bf = 0;
}
else if (bf == -1)
{
parent->_bf = 0;
subRL->_bf = 0;
subR->_bf = 1;
}
else
{
parent->_bf = -1;
subRL->_bf = 0;
subR->_bf = 0;
}
}
判断是否是平衡树：
bool _IsBalance(Node* root,int height)
{
if (root == NULL)
{
height = 0;
return true;
}
int l = 0;
int r = 0;
if (_IsBalance(root->_left, 1)&& _IsBalance(root->_right, r) && abs(r - 1) < 2)
{
height = l < r ? r + 1 : l + 1;
if (root->_bf != r - 1)
{
cout << "平衡因子异常"<_key <
下面是完整的代码：
#include
using namespace std;
#include
template
struct AVLTreeNode
{
AVLTreeNode* _left;
AVLTreeNode* _right;
AVLTreeNode* _parent;
K _key;
V _value;
int _bf;
AVLTreeNode(const K& key, const V& value)
:_left(NULL)
, _right(NULL)
, _parent(NULL)
, _bf(0)
, _key(key)
, _value(value)
{}
};
template
class AVLTree
{
typedef AVLTreeNode Node;
public:
AVLTree()
:_root(NULL)
{}
~AVLTree()
{}
bool Insert(const K& key, const V& value)
{
if (_root == NULL)
{
_root = new Node(key, value);
return true;
}
Node* parent = NULL;
Node* cur = _root;
//判断插入的值和cur的大小
while (cur)
{
parent = cur;
if (cur->_key < key)
{
cur = cur->_right;
}
else if (cur->_key>key)
{
cur = cur->_left;
}
else
{
return false;
}
}
cur = new Node(key, value);//构建一个新的节点
if (parent->_key > key)//如果父节点的值大于key,则key就为父节点的左边
{
parent->_left = cur;
cur->_parent = parent;
}
else // 如果父节点的值小于key, 则key就为父节点的右边
{
parent->_right = cur;
cur->_parent = parent;
}
while (parent)
{
if (cur == parent->_left)//如果插入的节点在parent的左边，则平衡因子-1
{
parent->_bf--;
}
else//如果插入的节点在parent的右边，则平衡因子+1
{
parent->_bf++;
}
if (parent->_bf == 0)//平衡因子为0，子树的高度没有变化
{
break;
}
else if(parent->_bf == -1 || parent->_bf == 1)
{
//子树的高度变化，但是高度没有变化，继续向上一层树进行检查，是否对上一层有影响
cur = parent;
parent = cur->_parent;
}
else//子树的高度为2或者-2 导致二叉树不再平衡，则需要进行旋转
{
if(parent->_bf==2)//说明右边子树的高度增加了
{
Node* subR = parent->_right;
if (subR->_bf == 1)
{
RotateL(parent);
}
else
{
RotateRL(parent);
}
}
else//说明左边的子树的高度增加了
{
Node* subL = parent->_left;
if (subL->_bf == -1)
{
RotateR(parent);
}
else
{
RotateLR(parent);
}
}
break;
}
}
}
void InOrder()
{
_InOrder(_root);
}
bool IsBalance()
{
int height = 0;
return _IsBalance(_root,height);
}
protected:
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)
{
subR = _root;
subR->_parent = NULL;
}
else
{
if (ppNode->_left == parent)
{
ppNode->_left = subR;
}
else
{
ppNode->_right = subR;
}
subR->_parent = ppNode;
}
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;
if (ppNode == NULL)
{
_root = subL;
subL->_parent = NULL;
}
else
{
if (ppNode->_left == parent)
{
ppNode->_left = subL;
}
else
{
ppNode->_right = subL;
}
subL->_parent = ppNode;
}
parent->_bf = subL->_bf = 0;
}
void RotateLR(Node* parent)
{
Node* subL = parent->_left;
Node* subLR = subL->_right;
int bf = subLR->_bf;
RotateL(subL);
RotateR(parent);
if (bf == 0)
{
parent->_bf = subL->_bf = subLR->_bf = 0;
}
else if (bf==1)
{
subL->_bf = -1;
parent->_bf = 0;
subLR->_bf = 0;
}
else
{
subL->_bf = 0;
subLR->_bf = 0;
parent->_bf = 1;
}
}
void RotateRL(Node* parent)
{
Node* subR = parent->_right;
Node* subRL = subR->_left;
int bf = subRL->_bf;
RotateR(subR);
RotateL(parent);
if (bf == 0)
{
parent->_bf = subR->_bf = subRL->_bf = 0;
}
else if (bf == -1)
{
parent->_bf = 0;
subRL->_bf = 0;
subR->_bf = 1;
}
else
{
parent->_bf = -1;
subRL->_bf = 0;
subR->_bf = 0;
}
}
void _InOrder(Node* root)//中序
{
if (root == NULL)
{
return;
}
_InOrder(root->_left);
cout << root->_key << ":" << root->_value << endl;
_InOrder(root->_right);
}
size_t _Height(Node* root)
{
if (root == NULL)
{
return 0;
}
size_t LHeight = _Height(root->_left);
size_t RHeight = _Height(root->_right);
return LHeight < RHeight ? RHeight + 1 : LHeight + 1;
}
bool _IsBalance(Node* root,int height)
{
if (root == NULL)
{
height = 0;
return true;
}
int l = 0;
int r = 0;
if (_IsBalance(root->_left, 1)&& _IsBalance(root->_right, r) && abs(r - 1) < 2)
{
height = l < r ? r + 1 : l + 1;
if (root->_bf != r - 1)
{
cout << "平衡因子异常"<_key < tree;
for (int i = 0; i < 9; i++)
{
tree.Insert(a[i], i);
}
tree.InOrder();
tree.IsBalance();
}
int main()
{
Test();
system("pause");
return 0;
}