Aspire Faculty ID #18540 · Topic: UGC NET Computer Science Dec 2022 Shift I (Paper II) · Just now
UGC NET Computer Science Dec 2022 Shift I (Paper II)

 Consider the following statements:

A. The number of labelled binary trees possible with $n$ nodes is $\frac{(2n)!}{(n+1)!}$

B. The number of binary search trees possible with $n$ nodes is $\frac{(2n)!}{(n+1)!n!}$

C. The number of unlabeled binary trees possible with $n$ nodes is $\frac{C(2n,n)}{(n+1)!}$

D. The number of simple, undirected and labelled graphs possible with $n$ vertices is $2^{\frac{n(n-1)}{2}}$

Solution

A is correct.

Number of labelled binary trees with $n$ nodes:

$=C_n\times n!$

$=\frac{(2n)!}{(n+1)!n!}\times n!$

$=\frac{(2n)!}{(n+1)!}$

B is correct.

Number of binary search trees with $n$ nodes:

$=C_n=\frac{(2n)!}{(n+1)!n!}$

C is incorrect.

Correct formula for unlabeled binary trees is:

$C_n=\frac{1}{n+1}C(2n,n)$

D is correct.

Number of simple undirected labelled graphs:

$=2^{\frac{n(n-1)}{2}}$

So, correct option is:

$\boxed{(b)\ \text{A, B and D are only correct}}$

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