平衡二叉树

平衡二叉树(AVL)是一种特殊的二叉搜索树，他满足两个性质：

1. 此树是二叉搜索树

2. 任意节点的左右子树高度差的绝对值不超过1

这样是为了提高查询的效率，因为一般的二叉搜索树有可能不会是完全二叉树或者接近完全二叉树情况，有的甚至退化成链表，所以平衡二叉树将二叉树平衡一下，使得查询效率满足logn，主要有四种情况，一种是LL型，RR型，LR型，RL型，具体见代码实现：

#include <iostream>
#include <algorithm>
#include <stdlib.h>
using namespace std;
typedef struct AVLTree{
    int data;
    struct AVLTree *lchild, *rchild;
    int height;
}AVLTree, *PAVLTree;
int getHeight(PAVLTree T)
{
    int height = 0;
    if (T != NULL)
    {
        height = 1;
        if (T->lchild != NULL)
            height = T->lchild->height + 1;
        if (T->rchild != NULL)
            height = max(height, T->rchild->height) + 1;
    }
    return height;
    //recursion version
//    if (T != NULL)
//        return max(getHeight(T->lchild), getHeight(T->rchild)) + 1;
//    else
//        return 0;
}
//LL type
PAVLTree SingleLeft(PAVLTree A)
{
    PAVLTree B = A->lchild;
    A->lchild = B->rchild;
    B->rchild = A;
    A->height = max(getHeight(A->lchild), getHeight(A->rchild)) + 1;
    B->height = max(getHeight(B->lchild), getHeight(B->rchild)) + 1;
    return B;
}
//RR type
PAVLTree SingleRight(PAVLTree A)
{
    PAVLTree B = A->rchild;
    A->rchild = B->lchild;
    B->lchild = A;
    //update height
    A->height = max(getHeight(A->lchild), getHeight(A->rchild));
    B->height = max(getHeight(B->lchild), getHeight(B->rchild));
    return B;
}
//LR type
PAVLTree Double_Left_Right(PAVLTree A)
{
    A->lchild = SingleRight(A->lchild);
    return SingleLeft(A);
}
//RL type
PAVLTree Double_Right_Left(PAVLTree A)
{
    A->rchild = SingleLeft(A->rchild);
    return SingleRight(A);
}
//insert function
PAVLTree insert_AVL_tree(int data, PAVLTree T)
{
    //the process same as binary search tree, but keep balance of the tree after inserted data
    if (T == NULL)
    {
        T = (PAVLTree)malloc(sizeof(AVLTree));
        T->data = data;
        T->lchild = T->rchild = NULL;
        T->height = 0;
    }
    if (data < T->data)
    {
        T->lchild = insert_AVL_tree(data, T->lchild);
        //if can not meeting the conditions
        if (getHeight(T->lchild) - getHeight(T->rchild) == 2)
        {
            if (data < T->lchild->data)
            {
                T = SingleLeft(T);
            }
            else
                T = Double_Left_Right(T);
        }
    }
    //if can not meeting the conditions
    else if (data > T->data)
    {
        T->rchild = insert_AVL_tree(data, T->rchild);
        if (getHeight(T->lchild) - getHeight(T->rchild) == -2)
        {
            if (data > T->rchild->data)
                T = SingleRight(T);
            else
                T = Double_Right_Left(T);
        }
    }
    //update the height
    T->height = max(getHeight(T->lchild), getHeight(T->rchild)) + 1;

    return T;


}
//previous order traverse
void preOrderTraverse(PAVLTree T)
{
    if (T != NULL)
    {
        cout << T->data << " ";
        preOrderTraverse(T->lchild);
        preOrderTraverse(T->rchild);
    }
}
int main()
{
    int n;
    cin >> n;//the number of test data
    int t;
    PAVLTree root = NULL;
    for (int i = 0; i < n; i++)
    {
        cin >> t;
        root = insert_AVL_tree(t, root);//create the AVL tree
    }
    preOrderTraverse(root);//show the result
    return 0;
}