It is well known 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 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 $n \to \infty$.