[It is well known](https://en.wikipedia.org/wiki/Cayley%27s_formula) that the number of labelled trees on $n$ vertices is equal to $n^{n-2}$.


1. We do not expect any such exact formula for [the number of isomorphism types](https://oeis.org/A000055) of trees on $n$ vertices. But what are the sharpest asymptotics, or best upper and lower bounds known, as $n \to \infty$?
2. Has anyone studied the number of homeomorphism types of trees on $n$ vertices? Again, I don't expect an exact answer, and am mostly interested in the asymptotics as $n \to \infty$.



How does the number of trees on $n$ vertices *up to isomorphism* grow as $n \to \infty$?
For Q1 the answer is known to be $\sim C\_1C\_2^n n^{-5/2} $ for $C\_1\approx 0.5349496061...$ and $C\_2\approx 2.9955765856...$. This can be found in Flajolet and Sedgewick's "Analytic Combinatorics" (see p.481) with the main ingredients being singularity analysis and the relation
$$I(z)=H(z)-\frac{1}{2}\left(H(z)^2-H(z^2)\right)$$
where $I$ is the generating function for unrooted unlabelled trees and $H$ is the generating function for rooted unlabelled trees. This is all originally from Otter's paper
"The Number of Trees" from Annals of Mathematics, Vol 49, no.3 pp. 583-599.


For Q2, homeomorphism classes of trees on $n$ vertices correspond to homeomorphically irreducible trees (also sometimes called series-reduced tress, or topological trees, see [OEIS](https://oeis.org/A000014)) of at most $n$ vertices. This enumeration problem appears in the movie Good Will Hunting and there is even a [numberphile video](https://www.youtube.com/watch?v=iW_LkYiuTKE) about it.


These were enumerated by Harary and Prins in



> 
> Frank Harary and Geert Prins, "The number of homeomorphically irreducible trees and other species" Acta Math., 101 (1959), 141-162
> 
> 
> 


and for the asymptotics see



> 
> F. Harary, R. W. Robinson and A. J. Schwenk, Twenty-step algorithm for determining the asymptotic number of trees of various species, J. Austral. Math. Soc., Series A, 20 (1975), 483-503
> 
> 
> 


I also believe that both Q1 and Q2 are exposited in the book "Graphical Enumeration" by Harary and Palmer, but I'll have to check.


