First of all, is this true? Since there is no closed formula for the number of unlabeled trees on n vertices it is not something I know how to prove using, for instance, induction. (This is an observation I made when writing out a few problems, but I'm not 100% sure if I can just assume it or there is something that needs to be proved first).
Might be a little bit of a naive question, but I am wondering if there is a combinatorial/intuitive explanation to this sort of question? Or is it something that follows trivially from the definition? That is, if we take the unlabeled trees on $n$ vertices and find the labelings for each one and then take the sum, it will basically give us Cayley's formula of distinct labelings on $n$ vertices.
Any help would be appreciated