[Solved]: Counting trees (order matters)

Problem Detail: 

As a follow up to this question (the number of rooted binary trees of size n), how many possible binary trees can you have if the nodes are now labeled, so that abc is different than bac cab etc ? In other words, order matters. Certainly it will be much more than the Catalan number.

What would the problem be if you have n-ary trees instead of binary ?

Are these known problems? reference ?

Asked By : user1419

Answered By : user1419

As partially answered above by Syzygy, for labelled binary trees, it is $n!C_n$, where $C_n$ being the Catalan number. Generalized this to labelled $k$-ary trees, it is $n!C^k_n$ where $C^k_n$ is the Fuzz-Catalan number $ \begin{equation} C^k_n= \binom{kn}{n}\frac{1}{(k-1)n+1} \end{equation} $

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Question Source : http://cs.stackexchange.com/questions/1762