No of binary tree

Question

number of distinct binary tree,that can be created by 5 nodes(distinct)

Answer 1

$5$ distinct nodes means you can uniquely identify each node based on some parameter say a key value, Then the solution to this problem is given by bell's number.

Number of binary trees possible $= \binom {2n}{n}*n!$

Comments

Could you please elaborate this solution numerically?

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I think its not correct.

#unlabelled binary trees possible = n^th catalan number

Now, being labelled , they cn be permuted in n! ways.

So, answer = $\frac{\binom{2n}{n}}{n+1} * n!$

Correct me if wrong.

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Bell no. gives partitions. It doesnt consider permutations.