GATE | GATE CS 2018 | Question 57

The number of possible min-heaps containing each value from {1, 2, 3, 4, 5, 6, 7} exactly once is _______.

Note –This was Numerical Type question.
(A) 80
(B) 8
(C) 20
(D) 210


Answer: (A)

Explanation: Set minimum element as root (i.e 1), now 6 are remaining and left subtree will have 3 elements, each left subtree combination can be permuted in 2! ways.

Total ways to design min-heap with 7-elements = *2! * 2! = 20*2*2 = 80

Alternative approach – Total number of min or max heap tree with 1 to N elements are using recurrence relation:

T(N) =(N-1)Ck * T(k) * T(N-k-1), where k = number of nodes on left subtree

T(1) = 1
T(2) = 1
T(3) = 2
T(4) = 3C2 * T(2) * T(1) = 3
T(5) = 4C3 * T(3) * T(1) = 8
T(6) = 5C3 * T(3) * T(2) = 20
T(7) = 5C3 * T(3) * T(3) = 80

So, answer is 80.

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