How many non-isomorphic, non-labelled, forests are there (asymptotically) on $n$ vertices?

Question

Is there any formula known? There is the following asympotic formula for unlabelled trees: $$t(n) \sim C \alpha^n n^{-\frac{5}{2}}$$ With $t(n)$ the number of unlabelled non-isomorphic trees on $n$ vertices. Is there a similar formula known for forests?

I actually just wanna know if the number of forests is $F(n)=O(\beta^n)$ for some constant $\beta$.

Answer 1

If all we want is a rough estimate, then it's enough to say that

1. $t(n)$ is a lower bound on $F(n)$, of course, but
2. $(n+1) t(n+1)$ is an upper bound on $F(n)$. If we take an $(n+1)$-vertex tree and delete a vertex from it, we get an $n$-vertex forest, and every forest can be obtained by doing this in at least one way.

The lower bound has order $\Theta(\alpha^n n^{-5/2})$, and the upper bound has order $\Theta(\alpha^n n^{-3/2})$, so $F(n)$ is somewhere in between.