Counting trees

Theorem [Cayley, 1889] The number of trees on n labeled vertices is n^{n-2}.

Theorem [Cayley, 1889] The number T_{n,k} of rooted forests on n labeled vertices with k trees is \binom{n-1}{k-1} n^{n-k}.

Borchardt (1860), Sylvester (1857) \sum_{k=1}^n T_{n,k} = (n + 1)^{n-1}.

The number of rooted forests on n labeled vertices with k trees using only the first k labels as roots is \frac{\binom{n-1}{k-1} n^{n-k}}{\binom{n}{k}} and \sum_{k=1}^n \frac{\binom{n-1}{k-1} n^{n-k}}{\binom{n}{k}} = \frac{n^n-n}{(n-1)^2}.

Problem list: