【PTA】【数据结构与算法】AVL树

判断题

1.Insert 1, 2, 3, 4, 5, and 6 one by one into an initially empty AVL tree. Then the preorder traversal sequence of the resulting tree must be {4, 2, 1, 3, 5, 6}. (2分)

T	F

解析：以下演示了插入的过程，最后旋转得到的AVL树通过前序遍历可得到答案。

2.For any node in an AVL tree, the height of the left subtree must be greater than that of the right subtree. (2分)

T	F

解析：AVL树的定义是任一结点左右子树的差至多为1，所以可以左子树的高度比右子树高一，也可以右子树的高度比左子树高一。

3.For an AVL tree, the balance factors of all the non-leaf nodes are 0 if the tree is a complete binary tree. (2分)

T	F

解析：完全二叉树不一定是满二叉树，可能有非叶子结点的左右子树高度不同。

4.If the depth of an AVL tree with nodes { 1, 2, 3, 4 } is 3 (the depth of the root is 1), then either node 2 or node 3 must have two children. (2分)

T	F

解析：因为有4个结点，存在AVL树的调平机制，只可能存在一个根结点、左一结点和右二结点，或一个根结点、左二结点和右一结点，而因为二和三都是中间的结点，所以必有一个作为根节点且有两个孩子。

5.任何AVL树的中序遍历结果是有序的（从小到大）。 (2分)

T	F

解析：AVL树是二叉搜索树，二叉搜索树的中序遍历是有序的（从小到大），故AVL树也满足。

6.For any node in an AVL tree, the height of the right subtree must be greater than that of the left subtree. (2分)

T	F

解析：AVL树的定义是任一结点左右子树的差至多为1，所以可以左子树的高度比右子树高一，也可以右子树的高度比左子树高一。

7.An AVL tree with the balance factors of all the non-leaf nodes being 0 must be a perfect binary tree. (2分)

T	F

解析：完美二叉树任一结点左右子树高度都是相等的且平衡因子都为0。

8.If the depth of an AVL tree with nodes { 1, 2, 3, 4 } is 3 (the depth of the root is 1), then it is possible for node 4 to be the root. (2分)

T	F

解析：如果4作为根节点，其余三个结点都在4的左子树上，高度为2，而右子树高度为0，不平衡。

选择题

1.Which of the following is TRUE for any node in an AVL tree? (2分)

	选项
A	The left and right subtrees have the same height
B	The absolute value of the difference between the heights of left and right subtrees is no more than 1
C	The height of the left subtree is always greater than that of the right subtree
D	The height of the left subtree is always less than that of the right subtree

解析：由定义得，任意结点左右子树的高度差的绝对值不大于1。

2.Insert key 48 into the balanced binary tree shown by the figure. Then in the resulting balanced tree, the left- and right-child of key 37 are: (2分)

	选项
A	13 and 48
B	24 and 48
C	24 and 53
D	24 and 90

解析：

3.Insert 2, 1, 4, 5, 9, 3, 6, 7 into an initially empty AVL tree. Which one of the following statements is FALSE? (2分)

	选项
A	4 is the root
B	3 and 7 are siblings
C	2 and 6 are siblings
D	9 is the parent of 7

解析：

4.If the depth of an AVL tree is 5 (the depth of an empty tree is defined to be −1), then the minimum possible number of nodes in this tree is: (2分)

	选项
A	12
B	20
C	33
D	64

解析：由递推公式N_h = N_h-1 +N_h-2 +1得，深度为5的AVL数最少结点是20（空树深度为-1）。

5.Insert { 9, 8, 7, 2, 3, 5, 6, 4} into an initially empty AVL tree. Which one of the following statements is FALSE? (2分)

	选项
A	5 is the root
B	2 and 5 are siblings
C	there are 2 nodes with their balance factors being -1
D	the height of the resulting AVL tree is 3

解析：

6.After inserting 28, 22, and 35 into the given AVL tree, the children of node 25 are __。 (2分)

	选项
A	22 and 28
B	20 and 28
C	22 and 30
D	20 and 30

解析：

7.Among the following trees, which one satisfies the definition of a balanced tree? (2分)

	选项
A
B
C
D

解析：由定义得，任意结点左右子树的高度差的绝对值不大于1。其余几个选项根结点就已经不满足定义了。

8.The maximum possible depth of an AVL tree with 12 nodes is: (2分)

	选项
A	3
B	4
C	5
D	6

解析：由递推公式N_h = N_h-1 +N_h-2 +1得，深度为5的AVL数最少结点是12（空树深度为0）。

9.Insert {88, 70, 61, 96, 120, 90} one by one into an initially empty AVL tree. Then the preorder traversal sequence of the resulting AVL tree is: (2分)

	选项
A	61,70,88,90,96,120
B	90,70,61,88,96,120
C	88,70,61,90,96,120
D	88,70,61,96,90,120

解析：

10.If the depth of an AVL tree is 6 (the depth of an empty tree is defined to be -1), then the minimum possible number of nodes in this tree is: (2分)

	选项
A	13
B	17
C	20
D	33

解析：

11.如果AVL树的深度为5（空树的深度定义为−1），则此树最少有多少个结点？(2分)

	选项
A	12
B	20
C	33
D	64

解析：由递推公式N_h = N_h-1 +N_h-2 +1得，深度为5的AVL数最少结点是20（空树深度为-1）。

12.将一系列数字顺序一个个插入一棵初始为空的AVL树。下面哪个系列的第一次旋转是“右-左”双旋？(2分)

	选项
A	1,2,3,4,5,6
B	6,5,4,3,2,1
C	4,2,5,6,3,1
D	3,1,4,6,5,2

解析：

13.将 1, 2, 3, 6, 5, 4 顺序一个个插入一棵初始为空的AVL树，会经历下列哪些旋转？ (2分)

	选项
A	两个“右-右”旋和一个“右-左”旋
B	一个“右-右”旋、一个“右-左”旋、一个“左-右”旋
C	一个“右-右”旋和两个“右-左”旋
D	两个“右-右”旋和一个“左-右”旋

解析：

14.在任意一棵非空平衡二叉树（AVL 树）T​₁​​中，删除某结点 v 之后形成平衡二叉树 T​₂​​，再将 v 插入 T​​₂​​形成平衡二叉树 T​​₃。下列关于 T​₁与 T​₃​​的叙述中，正确的是：(2分)

 I、若 v 是 T​​₁的叶结点，则 T​​₁与 T​​₃可能不相同
 II、若 v 不是 T​₁​​的叶结点，则 T​₁​​与 T​₃​​一定不同
 III、若 v 不是 T​₁​​的叶结点，则 T​₁​​与 T​₃​​一定相同

	选项
A	仅 I
B	仅 II
C	仅 I、II
D	仅 I、III

解析：

15.首先将 28, 23, 54, 61, 98, 37 插入一棵初始为空的平衡二叉树（AVL树），然后马上插入下列选项中的一个键值。哪个键值将引起 RL 旋转？

	选项
A	10
B	30
C	60
D	70

解析：

16.将 26, 13, 44, 51, 98, 37, 66, 73 顺序插入一棵初始为空的AVL树。下列句子中哪句是错的？

	选项
A	44 是根结点
B	37 和 73 是兄弟
C	26 和 66 是兄弟
D	26 是 13 的父结点

解析：