In mathoverflow posting 43081, the question was raised of bounding the number of spanning trees of a graph in terms of $m, n$, the number of edges and vertices. I am interested in the case when $m$ is small compared to $n$ (say $m = c n$ for $c \rightarrow 1$.). In this case, the best bound I have been able to find is the obvious bound $$ \kappa \leq \binom{m}{n-1} $$

There are a lot of papers describing tighter bounds when $m$ is large, such as the bound $\kappa < (2 m/n)^{n-1}$. But are there are any better bounds available in the regime when $m$ is small?

Thanks for any pointers!