PAT1066(AVL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 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
给出一个插入的序列，最后输出AVL树的根
主要就是针对AVL树的失衡的几种情况，分别进行重新平衡
LL型：由于在A左子树根节点的左子树上插入结点，使A的平衡由1增至2.此时可以通过右旋来实现。所谓右旋就是将根节点作为其左子树的右子树。
若原左子树上有右结点，此时作为根节点的左子树。
RR型：由于在A右子树根节点的右子树上插入节点，使A的平衡由-1变成-2.此时可以通过左旋实现。所谓左旋就是将根节点作为其右子树的左子树。
若原右子树有左孩子，此时作为根节点的右孩子。
LR型：由于在A左子树根节点的右子树上插入节点，使A的平衡由1增至2.此时可现对root->left进行一次左旋，再对root进行一次右旋
RL型：由于在A右子树根节点的左子树上插入节点，使A的平衡由-1变成-2.此时可现对root->right进行一次右旋，在对root进行一次左旋。
具体代码如下：

#include<iostream>
#include<cstdio>
using namespace std;
struct Node
{
    int val;
    Node *left,*right;
};
Node* rotateright(Node *root) //右旋 LL型
{
    Node *t=root->left;
    root->left=t->right;
    t->right=root;
    return t;
}
Node* rotateleft(Node *root) //左旋 RR型
{
    Node *t=root->right;
    root->right=t->left;
    t->left=root;
    return t;
}
Node* rotateleftright(Node *root) //LR型
{
    root->left=rotateleft(root->left);
    return rotateright(root);
}
Node* rotaterightleft(Node *root)
{
    root->right=rotateright(root->right);
    return rotateleft(root);
}
int getHeight(Node *root)
{
    if(root==NULL)
        return 0;
    else
        return max(getHeight(root->left),getHeight(root->right))+1;
}
Node* build(Node *root,int val)
{
    if(root==NULL)
    {
        root=new Node();
        root->val=val;
        root->left=root->right=NULL;
    }
    else if(val<root->val)//插入到左子树中
    {
        root->left=build(root->left,val);
        if(getHeight(root->left)-getHeight(root->right)==2)
        {
            root=val<root->left->val ?rotateright(root):rotateleftright(root); //如果是LL型，做右旋，否则先左旋后右旋
        }
    }
    else
    {
        root->right=build(root->right,val);
        if(getHeight(root->left)-getHeight(root->right)==-2)
            root=val>root->right->val ? rotateleft(root):rotaterightleft(root);
    }
    return root;
}
int main()
{
    int n;
    scanf("%d",&n);
    Node *root=NULL;
    for(int i=1;i<=n;i++)
    {
        int val;
        scanf("%d",&val);
        root=build(root,val);
    }
    printf("%d
",root->val);
}