04-树4. Root of AVL Tree

An AVL tree is a self-balancing binary search tree. In an AVL tree, the heights of the two child subtrees of any node differ by at most one; if at any time they differ by more than one, rebalancing is done to restore this property. Figures 1-4 illustrate the
rotation rules.


    



    


Now given a sequence of insertions, you are supposed to tell the root of the resulting AVL tree.

Input Specification:

Each input file contains one test case. For each case, the first line contains a positive integer N (<=20) which is the total number of keys to be inserted. Then N distinct integer keys are given in the next line. All the numbers in a line are separated by
a space.

Output Specification:

For each test case, print ythe root of the resulting AVL tree in one line.
Sample Input 1:

5
88 70 61 96 120

Sample Output 1:

70

Sample Input 2:

7
88 70 61 96 120 90 65

Sample Output 2:

88

这题主要考查了平衡二叉树的插入操作：

1. RR旋转、LL旋转比较好理解；

2. RL旋转、LR旋转则可以利用RR旋转和LL旋转操作得到：

若是需要LR旋转，则对T->left进行一次RR旋转，然后T就变成需要LL旋转，于是再进行一次LL旋转即可。（表达得是不是很清楚）

#include <iostream>

using namespace std;

typedef struct Node
{
int data;
Node* left;
Node* right;
int height;
Node(int d):data(d),left(NULL),right(NULL),height(0){};
}Node,*TreeRoot;

int Get_max(int a, int b)
{
return (a > b ? a : b);
}

//get the height of the tree or subtree
int Get_Height(TreeRoot T)
{
if(!T)
return -1;
return T->height;
}

TreeRoot RR_Rotation(TreeRoot A)
{
Node* B = A->right;
A->right = B->left;
B->left = A;

A->height = Get_max(Get_Height(A->left), Get_Height(A->right)) + 1;
B->height = Get_max(Get_Height(B->left), Get_Height(B->right)) + 1;

return B;
}

TreeRoot LL_Rotation(TreeRoot A)
{
Node* B = A->left;
A->left = B->right;
B->right = A;

A->height = Get_max(Get_Height(A->left), Get_Height(A->right)) + 1;
B->height = Get_max(Get_Height(B->left), Get_Height(B->right)) + 1;

return B;
}

TreeRoot RL_Rotation(TreeRoot A)
{
A->right = LL_Rotation(A->right);
return RR_Rotation(A);
}

TreeRoot LR_Rotation(TreeRoot A)
{
A->left = RR_Rotation(A->left);
return LL_Rotation(A);
}

TreeRoot AVLTree_Insert(int x, TreeRoot T)
{
if(!T)	//If the thee is empty.
{
T = new Node(x);
}
//If th
4000
e element>T->data, insert it at the right of the root
else if(x > T->data)
{
T->right = AVLTree_Insert(x, T->right);
if(Get_Height(T->right) - Get_Height(T->left) == 2)
{
if(x > T->right->data)
T = RR_Rotation(T);
else
T = RL_Rotation(T);
}
}
//If the element<T->data, insert it at the left of the root
else if(x < T->data)
{
T->left = AVLTree_Insert(x, T->left);
if(Get_Height(T->left) - Get_Height(T->right) == 2)
{
if(x < T->left->data)
T = LL_Rotation(T);
else
T = LR_Rotation(T);
}
}

else
return T;

T->height = Get_max(Get_Height(T->left), Get_Height(T->right)) + 1;

return T;
}

int main()
{
int n;
int input;
TreeRoot T = NULL;

cin>> n;

for(int i=0; i< n; i++)
{
cin>> input;
T = AVLTree_Insert(input, T);
}

cout<< T->data <<endl;

system("pause");
return 0;
}