PAT 1123 Is It a Complete AVL Tree

Posted 2020-11-20 mr-stn

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 output the level-order traversal sequence of the resulting AVL tree, and to tell if it is a complete binary tree.

Input Specification:

Each input file contains one test case. For each case, the first line contains a positive integer N (≤ 20). 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, insert the keys one by one into an initially empty AVL tree. Then first print in a line the level-order traversal sequence of the resulting AVL tree. All the numbers in a line must be separated by a space, and there must be no extra space at the end of the line. Then in the next line, print YES if the tree is complete, or NO if not.

Sample Input 1:

5
88 70 61 63 65

Sample Output 1:

70 63 88 61 65
YES

Sample Input 2:

8
88 70 61 96 120 90 65 68

Sample Output 2:

88 65 96 61 70 90 120 68
NO

题目大意：构建一颗avl，并且判断是否是完全二叉树， 输出层序遍历；
尝试用了算法笔记上面的方法， 代码的思路更加清晰一些， 构造的时间比以前的方法短一些； 
区别在于，在节点中添加了记录节点高度的变量， 在求节点高度的时候，不用递归的去求高度
         引用传值，也让代码简洁很多；
         在左旋右旋的过程中跟新节点高度；
如何证明二叉树是完全二叉树的方法见：https://www.cnblogs.com/mr-stn/p/9232618.html
注意点：在c++中0被认作是false，其他的值认为是true；

  1 #include<iostream>
  2 #include<vector>
  3 #include<queue>
  4 #include<algorithm>
  5 using namespace std;
  6 struct node{
  7   int val, height;
  8   node *left, *right;
  9 };
 10 
 11 int max(int a, int b){
 12     if(a>b) return a;
 13     else return b;
 14 }
 15 int getHeight(node* root){
 16   if(root==NULL) return 0;
 17   return root->height;
 18 }
 19 
 20 
 21 void updateHeight(node* &root){
 22     root->height = max(getHeight(root->left) , getHeight(root->right)) + 1;  
 23 }
 24 
 25 void leftRoate(node* &root){
 26   node* temp=root->right;
 27   root->right=temp->left;
 28   temp->left=root;
 29   updateHeight(root);
 30   updateHeight(temp);
 31   root = temp;
 32 }
 33 
 34 void rightRoate(node* &root){
 35   node* temp=root->left;
 36   root->left = temp->right;
 37   temp->right=root;
 38   updateHeight(root);
 39   updateHeight(temp);
 40   root = temp;
 41 }
 42 
 43 
 44 int getBalanceFactor(node* root){
 45   return getHeight(root->left) - getHeight(root->right);
 46 }
 47 
 48 
 49 void insert(node* &root, int val){
 50   if(root==NULL){
 51     root = new node;
 52     root->height=1;
 53     root->val = val;
 54     root->left = root->right=NULL;
 55     return ;
 56   }else if(root->val > val){
 57     insert(root->left, val);
 58     updateHeight(root);
 59     if(getBalanceFactor(root)==2){
 60       if(getBalanceFactor(root->left)==1) rightRoate(root);
 61       else{
 62         leftRoate(root->left);
 63         rightRoate(root);
 64       }
 65     }
 66   }else{
 67     insert(root->right, val);
 68     updateHeight(root);
 69     if(getBalanceFactor(root)==-2){
 70       if(getBalanceFactor(root->right)==-1) leftRoate(root);
 71       else{
 72         rightRoate(root->right);
 73         leftRoate(root);
 74       }
 75     }
 76   }
 77 }
 78 
 79 int main(){
 80   int n, i;
 81   cin>>n;
 82   vector<int> v(n);
 83   for(i=0; i<n; i++) scanf("%d", &v[i]);
 84   node *root=NULL;
 85   for(i=0; i<n; i++) insert(root, v[i]);
 86   queue<node*> q;
 87   q.push(root);
 88   int flag=0, f=0;
 89   vector<int> ans;
 90   while(q.size()){
 91     node* temp=q.front();
 92     q.pop();
 93     if(temp) ans.push_back(temp->val);
 94     if(temp->left != NULL){
 95       q.push(temp->left);
 96       if(flag) f=1;
 97     }else flag=true;
 98     if(temp->right != NULL){
 99       q.push(temp->right);
100       if(flag) f=1;
101     }else flag=true;
102   }
103   cout<<ans[0];
104   for(i=1; i<ans.size(); i++) cout<<" "<<ans[i];
105   if(f) cout<<endl<<"NO";
106   else cout<<endl<<"YES";
107   return 0;
108 }