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Data Structures and Algorithms Multiple Choice Questions
SET 08
1. What is the minimum number of nodes in an AVL tree of height 5?
a) 20
b) 21
c) 63
d) 12
Answer: (a) 20 Minimum number of nodes in an AVL tree can be recursively calculated as follows;Min. number of nodes of height h, N(h) = N(h-1) + N(h-2) + 1 Here, 1 is added to include the root node. N(0) = 1, N(1) = 2, N(2) = N(1) + N(0) + 1 = 2+1+1 = 4, N(3) = N(2) + N(1) + 1 = 4 + 2 + 1 = 7, N(4) = 12, N(5) = 20. AVL tree An AVL tree is a binary search tree with a self-balancing condition stating that the difference between the height of the left and right subtrees cannot be no more than one. An AVL tree has the following properties: 1. The sub-trees of every node differ in height by at most one. 2. Every sub-tree is an AVL tree. |
2. What is the minimum number of nodes that must be examined in order to find the minimum value in an AVL tree of height 5?
a) 20
b) 3
c) 12
d) 5
Answer: (b) 3 Minimum value in an AVL tree can be found at the left most node that has left pointer NULL. Such a node can be found at a height of 3 for an AVL tree of height 5. |
3. Consider a perfect binary tree of height h. Which of the following options represents the minimum and maximum number of nodes in the right sub-tree of the root?
a) 2^{h}, 2^{h}
b) 2^{h}-1, 2^{h}-1
c) 2^{h-1}, 2^{h-1}
d) 2h-1, 2h-1
Answer: (b) 2^{h}-1, 2^{h}-1 A perfect binary tree is a binary tree in which all interior nodes have two children and all leaves have the same depth or same level. In this tree each level contains the maximum number of nodes, ie., every level is completely full of nodes.The number of nodes in a perfect binary tree = 2^{h+1}-1 As per the question, we need to find the number of nodes in the right sub-tree alone. Also, the right sub-tree is at height less than that of original tree. Hence, the number of nodes = 2^{h}-1. For a perfect binary tree, the minimum and maximum number of nodes will be 2^{h+1}-1. Hence, for the right sub-tree alone, they are 2^{h}-1. |
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