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:

588 70 61 96 120

Sample Output 1:

70

Sample Input 2:

788 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 subtreeint 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 the element>T->data, insert it at the right of the rootelse 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);elseT = RL_Rotation(T);}}//If the element<T->data, insert it at the left of the rootelse 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);elseT = LR_Rotation(T);}}elsereturn 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;}