1123 Is It a Complete AVL Tree (30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 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

Output 1:

70 63 88 61 65
YES

##Sample

Input 2:

8
88 70 61 96 120 90 65 68

Output 2:

88 65 96 61 70 90 120 68
NO

Solution

主要分两步:

1. 建立avl树 (过程参见 AvlTree)
2. 进行层序遍历，遍历过程中判断是否为完全二叉树
3. 完全二叉树的定义为 最多有一个只有一个叶子的节点

Code

#include <iostream>
#include<cmath>
#include <queue>
using namespace std;
 

struct avl_node
{
    struct avl_node *left, *right;
    int data, height;
};
typedef struct avl_node *AvlTree, *AvlNode;

int height(AvlNode k)
{
    return k ? k->height : -1;
}

// lr 旋转
AvlNode left_right_rotate(AvlNode k2)
{
    AvlNode k1 = k2->left;
    k2->left = k1->right;
    k1->right = k2;

    k2->height = max(height(k2->left), height(k2->right)) + 1;
    k1->height = max(height(k1->left), height(k1->right)) + 1;
    return k1;
}

// rl 旋转
AvlNode right_left_rotate(AvlNode k1)
{
    AvlNode k2 = k1->right;
    k1->right = k2->left;
    k2->left = k1;

    k1->height = max(height(k1->left), height(k1->right)) + 1;
    k2->height = max(height(k2->left), height(k2->right)) + 1;
    return k2;
}

// ll 旋转
AvlNode left_left_rotate(AvlNode k3)
{
    k3->left = right_left_rotate(k3->left);
    return left_right_rotate(k3);
}

// rr 旋转
AvlNode right_right_rotate(AvlNode k1)
{
    k1->right = left_right_rotate(k1->right);
    return right_left_rotate(k1);
}

AvlNode insert(AvlNode t, int val)
{
    if (!t)
    {
        t = new avl_node();
        t->data = val;
        t->height = 0;
        t->left = t->right = nullptr;
        return t;
    }
    // self-check
    if (t->data > val)
    {
        t->left = insert(t->left, val);
        if (height(t->left) - height(t->right) == 2)
        {
            // lr 旋转
            if (t->left->data > val)
                t = left_right_rotate(t);
            else
                t = left_left_rotate(t);
        }
    }
    else
    {
        t->right = insert(t->right, val);
        if (height(t->right) - height(t->left) == 2)
        {
            // rl 旋转
            if (t->right->data < val)
                t = right_left_rotate(t);
            else
                t = right_right_rotate(t);
        }
    }
    t->height = max(height(t->left), height(t->right)) + 1;
    return t;
}

bool level_loop(AvlNode tree)
{
    queue<AvlNode> q;
    q.push(tree);
    int flag = 1;
    bool complete = true, leaf = false, falg = true;
    while (!q.empty())
    {
        AvlNode node = q.front();
        q.pop();
        if (!flag)
            cout << " ";
        else
            flag = false;
        cout << node->data;
        if (node->left)
            q.push(node->left);
        if (node->right)
            q.push(node->right);
        if (!node->left && node->right)
            complete = false;
        if (leaf && (node->left || node->right))
            complete = false;
        if (!node->left || !node->right)
            leaf = true;
    }
    cout << endl;
    return complete;
}

void pat_1123()
{

    int n, i, val;
    cin >> n;
    AvlTree root = nullptr;
    for (i = 0; i < n; i++)
    {
        cin >> val;
        root = insert(root, val);
    }
    bool complete = level_loop(root);
    if (complete)
        cout << "YES";
    else
        cout << "NO";
}

int main()
{
    pat_1123();
    return 0;
}