題目

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 the 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

分析

題目大意就是按順序插入平衡二叉樹，返回該 AVL 樹的根結點值 資料結構（六）平衡二叉樹 看完你就懂了

#include<iostream>
#include<malloc.h>
typedef struct AVLNode *AVLTree;
struct AVLNode{
	int data;     // 存值 
	AVLTree left;
 
  // 左子樹 
	AVLTree right;  // 右子樹 
	int height;  // 樹高 
};
using namespace std;

// 返回最大值 
int Max(int a,int b){
	return a>b?a:b;
}

// 返回樹高，空樹返回 -1 
int getHeight(AVLTree A){
	return A==NULL?-1:A->height;
}

// LL單旋
// 把 B 的右子樹騰出來掛給 A 的左子樹，再將 A 掛到 B 的右子樹上去 
AVLTree LLRotation(AVLTree A){
	// 此時根節點是 A 
	AVLTree B = A->left;  // B 為 A 的左子樹  
	A->left = B->right;   // B 的右子樹掛在 A 的左子樹上 
	B->right = A;     //  A 掛在 B 的右子樹上 
	A->height = Max(getHeight(A->left),getHeight(A->right)) + 1;
	B->height = Max(getHeight(B->left),A->height) + 1;
	return B;  // 此時 B 為根結點了 
}

// RR單旋
AVLTree RRRotation(AVLTree A){
	// 此時根節點是 A 
	AVLTree B = A->right;
	A->right = B->left;
	B->left = A;
	A->height = Max(getHeight(A->left),getHeight(A->right)) + 1;
	B->height = Max(getHeight(B->left),A->height) + 1;
	return B;  // 此時 B 為根結點了 
}

// LR雙旋 
AVLTree LRRotation(AVLTree A){
	// 先 RR 單旋
	A->left = RRRotation(A->left);
	// 再 LL 單旋 
	return LLRotation(A);
}

// RL雙旋
AVLTree RLRotation(AVLTree A){
	// 先 LL 單旋
	A->right = LLRotation(A->right);
	// 再 RR 單旋 
	return RRRotation(A); 
}

AVLTree Insert(AVLTree T,int x){
	if(!T){  // 如果該結點為空，初始化結點 
		T = (AVLTree)malloc(sizeof(struct AVLNode));
		T->data = x;
		T->left = NULL;
		T->right = NULL;
		T->height = 0;
	}else{  // 否則不為空， 
		if(x < T->data){  // 左子樹 
			T->left = Insert(T->left,x);
			if(getHeight(T->left)-getHeight(T->right)==2){  // 如果左子樹和右子樹高度差為 2 
				if(x < T->left->data)  // LL 單旋 
					T = LLRotation(T); 
				else if(T->left->data < x)  // LR雙旋
					T = LRRotation(T); 
			}
		}else if(T->data < x){
			T->right = Insert(T->right,x);
			if(getHeight(T->right)-getHeight(T->left)==2){
				if(x < T->right->data)  // RL 雙旋 
					T = RLRotation(T); 
				else if(T->right->data < x)  // RR單旋
					T = RRRotation(T); 
			}
		}
	}
	//更新樹高 
	T->height = Max(getHeight(T->left),getHeight(T->right)) + 1;
	return T;
} 


int main(){
	AVLTree T=NULL;
	int n;
	cin>>n;
	for(int i=0;i<n;i++){
		int tmp;
		cin>>tmp;
		T = Insert(T,tmp);
	}
	cout<<T->data;
	return 0;
}