Are there are any good lower bounds for the number of spanning trees for a *connected* graph $G$ in terms of (for example) number of edges $E$ or number of vertices $V$ ? 


Are improved bounds available if one knows the number of *bridges* of the graph? (A bridge, aka a cut-edge, is an edge whose removal disconnects the graph)


EDIT:


Here is a simpler statement of the question. What is the minimum number of spanning trees possible for a graph with $m$ edges and $n$ vertices which is (1) 1-edge-connected or (2) 2-edge-connected?



Lower bound on # spanning trees in a connected graph
A quick google turns up:


[Undirected simple connected graphs with minimum number of spanning trees](http://www.sciencedirect.com/science/article/pii/S0012365X08005050) by Zbigniew R. Bogdanowicz. According to this paper, the optimal graph is built as follows: Start with the complete graph $K\_{n-k}$. Take $k-1$ additional vertices and join them each to the complete graph by a single edge each. Finally, take one more vertex and join it to $m - \left( \binom{n-k}{2} + k-1 \right)$ of the vertices in the complete graph. (So we get $m$ edges total.) One should choose the parameter $k$ to be as small as possible, compatible with the requirement that $m - \left( \binom{n-k}{2} + k-1 \right)$ must be nonnegative.


I couldn't find any references on the $2$-connected version of this. 


