TITLE: Number of trees with n nodes and m leaves
QUESTION [9 upvotes]: Even searching for " 'number of trees' leaves " didn't reveal what I am looking for: an approach for calculating the (approximate) number of trees with exactly n nodes and m leaves. Any hints from MO?

REPLY [5 votes]: The number of labelled trees on $n$ vertices with $m$ leaves is 
$$\binom{n}{m}S(n-2,n-m)(n-m)!$$
where $S(a,b)$ is the Stirling number of the second kind. This can be seen by analysing the multivariate generation which counts trees of any degree sequence given here: https://math.berkeley.edu/~mhaiman/math172-spring10/matrixtree.pdf