1066 Root of AVL Tree (25分)

 

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 (≤) 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

平衡树模板题。

口诀：
先插入，后旋转。
插入记得更新高度。
旋转儿子父亲交换。
儿子的儿子也要管。
直线型儿子父亲交换。
S型先旋S下部变为直然后再旋直。

 1 #include <cstdio>
 2 #include <cstring>
 3 #include <cmath>
 4 using namespace std;
 5 
 6 struct Tree {
 7     int data, high;
 8     Tree *l, *r;
 9 } *root;
10 
11 int max(int a, int b) {
12     return a > b ? a : b;
13 }
14 
15 typedef Tree * ptree;
16 
17 int get_high(ptree root) {
18     if(root == NULL) return 0;
19     else return root -> high;
20 }
21 
22 void update_high(ptree root) {
23     root -> high = max(get_high(root -> l), get_high(root -> r)) + 1;
24 }
25 
26 int balance_factor(ptree root) {
27     return get_high(root -> l) - get_high(root -> r);
28 }
29 
30 void left_rotary(ptree &root) {
31     ptree temp = root -> r;
32     root -> r = temp -> l;
33     temp -> l = root;
34     update_high(root);
35     update_high(temp);
36     root = temp;
37 }
38 
39 //右旋：root做root的左儿子的右儿子，那么如果root的左儿子原本就有右儿子，需要把他变为root的左儿子。然后再将root变为他的左儿子的右儿子，记得更新这两颗子树的高度，还要将这两颗子树的
40 //根节点变为root的左儿子
41 
42 void right_rotary(ptree &root) {
43     ptree temp = root -> l;
44     root -> l = temp -> r;
45     temp -> r = root;
46     update_high(root);
47     update_high(temp);
48     root = temp;
49 }
50 
51 void avl_insert(ptree &root, int num) {
52     if(root == NULL) {//叶子节点，设定叶子节点高度为1
53         root = new Tree;
54         root -> data = num;
55         root -> high = 1;
56         root -> l = root -> r = NULL;
57     } else {
58         if(num < root -> data) {
59             avl_insert(root -> l, num);
60             update_high(root);//在root的子树中插入新元素需要重新调整这个树的高度
61             if(balance_factor(root) == 2) {//在root的子树中插入新元素后需要检查树是否平衡
62                 if(balance_factor(root -> l) == 1) {//如果root的左子树高度为1，则说明需要右旋
63                     right_rotary(root);
64                 } else if(balance_factor(root -> l) == -1) {//如果root的左子树比右子树低，则需要先左旋再右旋
65                     left_rotary(root -> l);
66                     right_rotary(root);
67                 } 
68             }
69         } else {
70             avl_insert(root -> r, num);
71             update_high(root);
72             if(balance_factor(root) == -2) {//如果是插入到root的右子树的话，那么root的右子树比左子树高2才需要调整
73                 if(balance_factor(root -> r) == -1) {
74                     left_rotary(root);
75                 } else if(balance_factor(root -> r) == 1) {
76                     right_rotary(root -> r);
77                     left_rotary(root);
78                 }
79             }
80         }
81     }
82 }
83 
84 int main() {
85     int n, num;
86     scanf("%d", &n);
87     while(n --) {
88         scanf("%d", &num);
89         avl_insert(root, num);
90     }
91     printf("%d\n", root -> data);
92     return 0;
93 }