GATE CS 2017 Set 1 — Question 6
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Programming & Data Structures → Trees → Binary Search Trees
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Question
Let be a binary search tree with 15 nodes. The minimum and maximum possible heights of are:
Note: The height of a tree with a single node is 0.
Note: The height of a tree with a single node is 0.
Correct answer
(B) 3 and 14 respectively
Solution
The height of a tree is defined as the number of edges on the longest path from the root to a leaf. For a single node, height is 0.Minimum Height:
A binary tree has minimum height when it is as complete as possible. The maximum number of nodes in a binary tree of height is given by .
We need
For , max nodes . Thus, a complete binary tree of height 3 has exactly 15 nodes.
Minimum height = 3.Maximum Height:
A binary tree has maximum height when it is a skewed tree (each node has only one child).
For nodes, the number of edges in a skewed tree is .
Maximum height .Therefore, the minimum and maximum heights are 3 and 14 respectively.
A binary tree has minimum height when it is as complete as possible. The maximum number of nodes in a binary tree of height is given by .
We need
For , max nodes . Thus, a complete binary tree of height 3 has exactly 15 nodes.
Minimum height = 3.Maximum Height:
A binary tree has maximum height when it is a skewed tree (each node has only one child).
For nodes, the number of edges in a skewed tree is .
Maximum height .Therefore, the minimum and maximum heights are 3 and 14 respectively.
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