AVL树_avl tree全称

AVL树全称高度平衡二叉树，它是一种优化了的搜索二叉树。我们知道，由于搜索二叉树存在缺陷，有可能退化成单链，这样的话搜索效率就降低了。为了将二叉搜索树的效率控制在O(logN)，所以对二叉搜索树加上一些条件，使得二叉搜索树高度平衡，时间复杂度为O(log N).

一   定义：

           1.首先AVL树是一个二叉搜索树

           2.它的左子树，右子数都是AVL树

           3.它的左子树和右子数的高度差不超过1.（通常情况下，为每个结点都加一个平衡因子bf，这个平衡因子是右子树的高度减去左子树高度）

二  平衡化旋转

如果一棵树原来是平衡的二叉搜索树，现在向里面添加一个结点。造成了不平衡。这时候我们就要调整这棵树的结构，使之重新平衡。以下是造成树不平衡的结构有哪些，以及如何调整。

下面关于平衡的几点，需要注意：

              1.所有的插入都是建立在平衡二叉树的基础上的

              2.如果插入一个结点，树的高度不变，则它就是平衡的

              3.插入之后会影响从插入点到根节点路径上结点的平衡因子

        

#include<iostream>
#include<assert.h>
using namespace std;
template<class V>
struct AVLTreeNode
{
	V _value;
	int _bf;//平衡因子右子树的高度减去左子树的高度
	AVLTreeNode< V>* _left;
	AVLTreeNode< V>* _right;
	AVLTreeNode< V>* _parent;
	AVLTreeNode( const V& value)
		:_value(value)
		, _bf(0)
		, _left(NULL)
		, _right(NULL)
		, _parent(NULL)
	{}
 
};
template<class V>
class AVLTree
{
	typedef AVLTreeNode< V> Node;
public:
	AVLTree()
		:_root(NULL)
	{}
	AVLTree(const AVLTree< V>& t)
	{
		_root = _Copy(t._root);
	}
 
	AVLTree< V>& operator=(const AVLTree< V>& t)//现代写法
	{
		if (this != &t)
		{
			AVLTree<K, V> tmp(t);
			swap(tmp._root, _root);
		}
		return *this;
	}
	/*AVLTree< V>& operator=(const AVLTree< V> t)
	{
		swap(_root, t._root);
		return *this;
	}*/
	~AVLTree()
	{
		_Destory(_root);
	}
	bool Isbalance()
	{
		//return _Isbalance(_root);
		int height = 0;
		return _IsbalanceOP(_root, height);
	}
	bool Insert( V& value)
	{
		Node* cur = _root;
		Node* parent = NULL;
		if (_root == NULL)
		{
			_root = new Node( value);
			return true;
		}
		while (cur)
		{
			if (cur->_value < value)
			{
				parent = cur;
				cur = cur->_right;
			}
			else if (cur->_value>value)
			{
				parent = cur;
				cur = cur->_left;
			}
			else
			{
				return false;
			}
		}
		 cur = new Node(value);
		if (parent->_value>value)
		{
			parent->_left = cur;
			cur->_parent = parent;
		}
		else
		{
			parent->_right = cur;
			cur->_parent = parent;
		}
		while (parent)//更新平衡因子
		{
			if (parent->_left ==cur)
			{
				parent->_bf--;
			}
			else
			{
				parent->_bf++;
			}
			if (parent->_bf == 0)//当父亲结点的平衡因子为0，说明插入之后树高度不变仍然平衡
			{
				return true;
			}
			else if (parent->_bf == 1 || parent->_bf == -1)//本次不违反规则，但是会影响根节点到parent路径上其他结点的平衡，因此继续向上更新
			{
				cur = parent;
				parent = parent->_parent;
			}
			else if (parent->_bf == 2 || parent->_bf == -2)//已经违反平衡规则，不再更新，直接旋转
			{
 
				if (parent->_bf == 2)
				{
					if (cur->_bf == 1)
					{
						_RotateL(parent);
						return true;
					}
					else
					{
						_RotateRL(parent);
						return true;
					}
				}
				if (parent->_bf == -2)
				{
					if (cur->_bf == -1)
					{
						_RotateR(parent);
						return true;
 
					}
					else
					{
						_RotateLR(parent);
						return true;
					}
				}
			}
 
		}
		return  true;
	}
	void _RotateR(Node* parent)
	{
		assert(parent);
		Node* SubL = parent->_left;
		Node* SubLR = SubL->_right;
		Node* ppNode = parent->_parent;
		parent->_left = SubLR;
		if (SubLR)
		{
			SubLR->_parent = parent;
		}
		SubL->_right = 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 _RotateL(Node *parent)
	{
		assert(parent);
		Node* SubR = parent->_right;
		Node* SubRL = SubR->_left;
		Node* ppNode = parent->_parent;
		parent->_right = SubRL;
		if (SubRL)
		{
			SubRL->_parent = parent;
		}
 
		SubR->_left = parent;
		parent->_parent = SubR;
		if (ppNode == NULL)
		{
			_root = SubR;
			SubR->_parent = NULL;
		}
		else
		{
			if (ppNode->_left == parent)
			{
				ppNode->_left = SubR;
			}
			else
			{
				ppNode->_right = SubR;
			}
			SubR->_parent = ppNode;
		}
		parent->_bf = SubR->_bf = 0;
 
	}
	void _RotateLR(Node *parent)
	{
		Node* SubL = parent->_left;
		Node* SubLR = SubL->_right;
		int bf = SubLR->_bf;//计算旋转以后的根节点的bf
		_RotateL(SubL);
		_RotateR(parent);
		//如果bf为0,代表插入点就是这个结点，这个时候所有旋转后bf皆为0
		if (bf == 0)
		{
			SubL->_bf = parent->_bf = 0;
			SubLR->_bf = 0;
		}
		//当bf为1时，这个时候相当于是在给bf的右树插入，
		else if (bf == -1)
		{
			SubL->_bf = -1;
			parent->_bf = 0;
			SubLR->_bf = 0;
		}
		//当bf为-1时，这时相当于给bf的左树进行插入
		else
		{
			SubL->_bf = 0;
			parent->_bf = 1;
			SubLR->_bf = 0;
		}
		
	}
	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 = 0;
			SubRL->_bf = 0;
		}
		else if (bf == 1)
		{
			parent->_bf = -1;
			SubR->_bf = 0;
			SubRL->_bf = 0;
		}
		else
		{
			parent->_bf = 0;
			SubR->_bf = 1;
			SubRL->_bf = 0; 
 
		}
	}
	size_t Height()
	{
		return _Height(_root);
	}
	void Inorder()
	{
		_Inorder(_root);
		cout << endl;
	}
protected:
	void _Inorder(Node* Root)
	{
		if (Root == NULL)
		{
			return;
		}
		else
		{
			_Inorder(Root->_left);
			cout << Root->_value<<" ";
			_Inorder(Root->_right);
		}
	}
	size_t _Height(Node* Root)
	{
		if (Root == NULL)
		{
			return 0;
		}
		else
		{
			int leftheight = _Height(Root->_left);
			int rightheight = _Height(Root->_right);
			return leftheight > rightheight ? leftheight + 1 : rightheight + 1;
		}
	}
	Node* _Copy(Node* Root)
	{
		Node* tmp = NULL;
		
		if (Root!=NULL)
		{
			tmp = new Node(Root->_value);
			tmp->_left = _Copy(Root->_left);
			tmp->_right = _Copy(Root->_right);
		}
		return tmp;
	}
	Node* _Destory(Node* Root)
	{
 
		if (Root != NULL)
		{
			Root->_left = _Destory(Root->_left);
			Root->_right = _Destory(Root->_right);
			delete Root;
			Root = NULL;
		}
	   return Root;
	}
	bool _Isbalance(Node* Root)//时间复杂度为n^2
	{
		if (Root == NULL)
		{
			return true;
		}
		int LeftHeight = _Height(Root->_left)+1;
		int RightHeight = _Height(Root->_right)+1;
		int sub = abs(RightHeight - LeftHeight);
		return ((sub < 2) && (_Isbalance(Root->_left)) && (_Isbalance(Root->_right)));
	}
	bool _IsbalanceOP(Node* Root, int& height)
	{
		if (Root == NULL)
		{
			height = 0;
			return true;
		}
		//记录左结点和右结点高度
		int LeftHeight = 0;
		int RightHeight = 0;
		if (_IsbalanceOP(Root->_left, LeftHeight) == false)
		{
			return false;
		}
		if (_IsbalanceOP(Root->_right, RightHeight) == false)
		{
			return false;
		}
		height = LeftHeight > RightHeight ? LeftHeight + 1 : RightHeight + 1;
		return abs(RightHeight - LeftHeight) < 2;
	}
private:
	Node* _root;
 
};
void TestAVLTree()
{
	//int a1[] = { 16, 3, 7, 11, 9, 26, 18, 14, 15 };
	int a1[] = { 4, 2, 6, 1, 3, 5, 15, 7, 16, 14 };
	AVLTree<int>t;
	for (size_t i = 0; i < sizeof(a1) / sizeof(a1[0]); i++)
	{
		t.Insert(a1[i]);
		cout <<a1[i]<<" "<< t.Isbalance() << endl;
	}
	t.Inorder();
	cout << t.Isbalance() << endl;
	AVLTree<int>t1(t);
	t1.Inorder();
	
 
}
int main()
{
	TestAVLTree();
	system("pause");
	return 0;
}