PAT.1066 Root of AVL Tree(平衡树模板题)

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。

#include <cstdio>
#include <cstring>
#include <cmath>
using namespace std;

struct Tree {
    int data, high;
    Tree *l, *r;
} *root;

int max(int a, int b) {
    return a > b ? a : b;
}

typedef Tree * ptree;

int get_high(ptree root) {
    if(root == NULL) return 0;
    else return root -> high;
}

void update_high(ptree root) {
    root -> high = max(get_high(root -> l), get_high(root -> r)) + 1;
}

int balance_factor(ptree root) {
    return get_high(root -> l) - get_high(root -> r);
}

void left_rotary(ptree &root) {
    ptree temp = root -> r;
    root -> r = temp -> l;
    temp -> l = root;
    update_high(root);
    update_high(temp);
    root = temp;
}

//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
//根节点变为root的左儿子

void right_rotary(ptree &root) {
    ptree temp = root -> l;
    root -> l = temp -> r;
    temp -> r = root;
    update_high(root);
    update_high(temp);
    root = temp;
}

void avl_insert(ptree &root, int num) {
    if(root == NULL) {//叶子节点，设定叶子节点高度为1
        root = new Tree;
        root -> data = num;
        root -> high = 1;
        root -> l = root -> r = NULL;
    } else {
        if(num < root -> data) {
            avl_insert(root -> l, num);
            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
            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
                if(balance_factor(root -> l) == 1) {//如果root的左子树高度为1，则说明需要右旋
                    right_rotary(root);
                } 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
                    left_rotary(root -> l);
                    right_rotary(root);
                } 
            }
        } else {
            avl_insert(root -> r, num);
            update_high(root);
            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
                if(balance_factor(root -> r) == -1) {
                    left_rotary(root);
                } else if(balance_factor(root -> r) == 1) {
                    right_rotary(root -> r);
                    left_rotary(root);
                }
            }
        }
    }
}

int main() {
    int n, num;
    scanf("%d", &n);
    while(n --) {
        scanf("%d", &num);
        avl_insert(root, num);
    }
    printf("%d\n", root -> data);
    return 0;
}