GATE CS 2016 Set 2 — Question 21
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Question
Breadth First Search (BFS) is started on a binary tree beginning from the root vertex. There is a vertex at a distance four from the root. If is the -th vertex in this BFS traversal, then the maximum possible value of is ______.
Correct answer
31 to 31
Solution
In a BFS traversal of a binary tree, nodes are visited level by level (distance from root).
To maximize , we assume the binary tree is a complete binary tree.
The number of nodes at distance is at most .
Nodes at distance 0:
Nodes at distance 1:
Nodes at distance 2:
Nodes at distance 3:
Total nodes at distances 0 to 3 .
The nodes at distance 4 will be visited after all nodes at distances 0 to 3.
The maximum number of nodes at distance 4 is .
The last node at distance 4 will be the -st node visited.
Thus, the maximum possible value of is 31.
To maximize , we assume the binary tree is a complete binary tree.
The number of nodes at distance is at most .
Nodes at distance 0:
Nodes at distance 1:
Nodes at distance 2:
Nodes at distance 3:
Total nodes at distances 0 to 3 .
The nodes at distance 4 will be visited after all nodes at distances 0 to 3.
The maximum number of nodes at distance 4 is .
The last node at distance 4 will be the -st node visited.
Thus, the maximum possible value of is 31.
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