PAT (Advanced Level) 1066. Root of AVL Tree (25) AVL树的插入建树

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

在插入一个新结点时，有四种情况会导致AVL树失去平衡，分别为左子树外侧，左子树内侧，右子树外侧，右子树内侧；外侧的情况比较简单，对根结点一次单旋转即可使树平衡，内侧的情况需要两次单旋转，首先将与根结点的值最接近的结点旋转为根结点的子结点，然后对根节点进行一次单旋转。

创建树时使用递归找到插入点，然后判断AVL树是否失去平衡（左右子树的高度差变为2），若失衡，判断是四种情况中的哪种，并作相应处理。

/*2015.7.28cyq*/
#include <iostream>
#include <vector>
using namespace std;

struct TNode{
int val;
TNode *left;
TNode *right;
TNode(int x):val(x),left(nullptr),right(nullptr){}
};

int height(TNode* &root){
if(root==nullptr)
return 0;
else
return max(height(root->left),height(root->right))+1;
}
TNode* singleRotateLeft(TNode* &root){//左子树外侧太深，单旋转
TNode* tmp=root->left;
root->left=tmp->right;
tmp->right=root;
return tmp;
}
TNode* singleRotateRight(TNode* &root){//右子树外侧太深，单旋转
TNode* tmp=root->right;
root->right=tmp->left;
tmp->left=root;
return tmp;
}
//双旋转时，先将待旋转子树中与root最接近的旋转为root的子结点，然后对root单旋转
TNode* doubleRotateLeft(TNode* &root){//左子树内侧太深，双旋转
root->left=singleRotateRight(root->left);
return singleRotateLeft(root);
}
TNode* doubleRotateRight(TNode* &root){//右子树内侧太深，双旋转
root->right=singleRotateLeft(root->right);
return singleRotateRight(root);
}
TNode* insert(TNode* root,int x){
if(root==nullptr)
root=new TNode(x);
else if(x< root->val){
root->left=insert(root->left,x);
if(height(root->left)-height(root->right)==2){
if(x< root->left->val)
root=singleRotateLeft(root);
else
root=doubleRotateLeft(root);
}
}else{
root->right=insert(root->right,x);
if(height(root->right)-height(root->left)==2){
if(x> root->right->val)
root=singleRotateRight(root);
else
root=doubleRotateRight(root);
}
}
return root;
}
int main(){
int N;
cin>>N;
TNode *root=nullptr;
int x;
while(N--){
cin>>x;
root=insert(root,x);
}
cout<<root->val;
return 0;
}