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GATE | GATE-CS-2015 (Set 2) | Question 20

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  • Difficulty Level : Basic
  • Last Updated : 28 Jun, 2021
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A binary tree T has 20 leaves. The number of nodes in T having two children is
(A) 18
(B) 19
(C) 17
(D) Any number between 10 and 20


Answer: (B)

Explanation:

Sum of all degrees = 2 * |E|.

Here considering tree as a k-ary tree :


Sum of degrees of leaves + Sum of degrees for Internal Node except root + Root's degree = 2 * (No. of nodes - 1).

Putting values of above terms,

L + (I-1)*(k+1) + k = 2 * (L + I - 1)

L + k*I - k + I -1 + k = 2*L + 2I - 2

L + K*I + I - 1 = 2*L + 2*I - 2

K*I + 1 - I = L

(K-1)*I + 1 = L

Given k = 2, L=20

==> (2-1)*I + 1 = 20
==> I = 19
==> T has 19 internal nodes which are having two children.

See Handshaking Lemma and Interesting Tree Properties for proof.

This solution is contributed by Anil Saikrishna Devarasetty

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