I hope I can avoid being confusing, but here goes.
I have a graph $(V, E)$, connected, undirected and with no loops. I also have an assignment of integer-valued weight to each edge of the graph. Note that this is the set up of the minimum spanning tree problem whose solution is well-know.
My question is: given an integer $n$, how many spanning trees are there of weight $n$? Is there a solution to compute this number? In other words, a function $f_{(V,E)}: \mathbb{N} \rightarrow \mathbb{N}$ that gives the number of spanning trees of a given weight. Perhaps something like Kirchoff's matrix tree theorem which gives you the total number of spanning trees.
THANKS!