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GATE | GATE CS 2018 | Question 57

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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:

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 =

^6C_3 *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) = 6C3 * T(3) * T(3) = 80

So, answer is 80.



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Last Updated : 09 Mar, 2018
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