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
题目大意：根据输入，建立平衡二叉树，输出根结点元素。
分析：熟悉四种调整方式。
参考代码：
#include<iostream>
#include<cstdlib>
using namespace std;
typedef int ElementType;
typedef struct AVLNode *Position;
typedef Position AVLTree; /* AVL树类型 */
struct AVLNode {
ElementType Data; /* 结点数据 */
AVLTree Left; /* 指向左子树 */
AVLTree Right; /* 指向右子树 */
int Height; /* 树高 */
};
AVLTree SingleLeftRotation(AVLTree A); /*LL旋转*/
AVLTree SingleRightRotation(AVLTree A); /*RR旋转*/
AVLTree DoubleLeftRightRotation(AVLTree A); /*LR旋转*/
AVLTree DoubleRightLeftRotation(AVLTree A); /*RL旋转*/
AVLTree Insert(AVLTree T, ElementType X);
int Max(int a, int b);
int GetHeight(AVLTree P);
void FreeTree(AVLTree T);
int main()
{
int N, i, v;
AVLTree T = NULL;
cin >> N;
for (i = 1; i <= N; i++)
{
cin >> v;
T=Insert(T,v);
}
cout << T->Data;
FreeTree(T);
return 0;
}
AVLTree SingleLeftRotation(AVLTree A)
{ /* 注意：A必须有一个左子结点B */
/* 将A与B做左单旋，更新A与B的高度，返回新的根结点B */
AVLTree B = A->Left;
A->Left = B->Right;
B->Right = A;
A->Height = Max(GetHeight(A->Left), GetHeight(A->Right)) + 1;
B->Height = Max(GetHeight(B->Left), A->Height) + 1;
return B;
}
AVLTree SingleRightRotation(AVLTree A)
{
AVLTree B = A->Right;
A->Right = B->Left;
B->Left = A;
A->Height = Max(GetHeight(A->Left), GetHeight(A->Right)) + 1;
B->Height = Max(GetHeight(B->Left), A->Height) + 1;
return B;
}
AVLTree DoubleLeftRightRotation(AVLTree A)
{ /* 注意：A必须有一个左子结点B，且B必须有一个右子结点C */
/* 将A、B与C做两次单旋，返回新的根结点C */
/* 将B与C做右单旋，C被返回 */
A->Left = SingleRightRotation(A->Left);
/* 将A与C做左单旋，C被返回 */
return SingleLeftRotation(A);
}
AVLTree DoubleRightLeftRotation(AVLTree A)
{
/*将A，B与C做两次单旋，返回新的根结点C*/
/*B与C做左单旋，C被返回*/
A->Right = SingleLeftRotation(A->Right);
/*将A与C做右单旋，C被返回*/
return SingleRightRotation(A);
}
AVLTree Insert(AVLTree T, ElementType X)
{ /* 将X插入AVL树T中，并且返回调整后的AVL树 */
if (!T) { /* 若插入空树，则新建包含一个结点的树 */
T = (AVLTree)malloc(sizeof(struct AVLNode));
T->Data = X;
T->Height = 0;
T->Left = T->Right = NULL;
} /* if (插入空树) 结束 */
else if (X < T->Data) {
/* 插入T的左子树 */
T->Left = Insert(T->Left, X);
/* 如果需要左旋 */
if (GetHeight(T->Left) - GetHeight(T->Right) == 2)
if (X < T->Left->Data)
T = SingleLeftRotation(T); /* 左单旋 */
else
T = DoubleLeftRightRotation(T); /* 左-右双旋 */
} /* else if (插入左子树) 结束 */
else if (X > T->Data) {
/* 插入T的右子树 */
T->Right = Insert(T->Right, X);
/* 如果需要右旋 */
if (GetHeight(T->Left) - GetHeight(T->Right) == -2)
if (X > T->Right->Data)
T = SingleRightRotation(T); /* 右单旋 */
else
T = DoubleRightLeftRotation(T); /* 右-左双旋 */
} /* else if (插入右子树) 结束 */
/* else X == T->Data，无须插入 */
/* 别忘了更新树高 */
T->Height = Max(GetHeight(T->Left), GetHeight(T->Right)) + 1;
return T;
}
int GetHeight(AVLTree T)
{
if (T == NULL)
return 0;
else return T->Height;
}
int Max(int a, int b)
{
return a > b ? a : b;
}
void FreeTree(AVLTree T)
{
if (T != NULL)
{
if (T->Left)FreeTree(T->Left);
if (T->Right)FreeTree(T->Right);
free(T);
}
}