【PAT（甲级）】1066 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 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

解题思路：

题目意思很清楚了，就是给你一个数组，让你构造平衡二叉搜索树。这里就涉及到树的旋转问题，关于树的旋转分为4类分别为LL，RR，LR，RL；对于后面两种，分别可以先对它的子树进行RR或先进行LL，转换后变成LL或RR类型，所以我主要还是讲一下LL和RR的旋转方法。

LL类就是在当前根节点的左子树的左子树上插入了新的节点，导致树不平衡，需要旋转来构造新的平衡。例如原题中的这张图就是LL类型（图中左数的深度为2，右数的深度为0，发生了不平衡）：

对于此，我们的做法是：此时不平衡的点权重为88记作root；

t = root->left; root->left = t->right; t->right = root; return t;

意思就是，把左树的节点作为根节点，左树的左树为根的左树，左树的右树，为root的左树

右树类似。

易错点：

只有读取入新的节点的时候才有可能会造成原来树的不平衡，所以只要在插入新节点的时候

代码：

#include
using namespace std;
 
typedef struct node Tree;
struct node{
	int node;
	Tree* left;
	Tree* right;
};
 
Tree* LL(Tree* root){
	Tree* t = root->left;
	root->left = t->right;
	t->right = root;
	return t;
}
 
Tree* RR(Tree* root){
	Tree* t = root->right;
	root->right = t->left;
	t->left = root;
	return t;
}
 
Tree* LR(Tree* root){
	root->left = RR(root->left);
	return LL(root);
}
 
Tree* RL(Tree* root){
	root->right = LL(root->right);
	return RR(root);
}
 
int getHeight(Tree* root){
	if(root == NULL){
		return 0;
	}
	return max(getHeight(root->left),getHeight(root->right))+1;
}
 
void read(Tree* &root, int val){
	if(root == NULL){
		root = new Tree();
		root->left = NULL;
		root->right = NULL;
		root->node = val;	
	}else if(val < root->node){
		read(root->left,val);
		if(getHeight(root->left)-getHeight(root->right) == 2){
			root = val < root->left->node ? LL(root):LR(root);
		}
    }
    else{
		read(root->right,val);
		if(getHeight(root->left)-getHeight(root->right) == -2){
			root = val > root->right->node ? RR(root):RL(root);
		}
	}
 
}
 
int main(){
	int N;
	cin>>N;
	Tree* tree = NULL;
	for(int i=0;i
		int t;
		cin>>t;
		read(tree,t);
	}
	cout<node;
	return 0;
}