Total number of spanning trees of a set of graphs
Can't this problem be used to compute the permanent of a 0-1 matrix? Since permanent is a #P-complete problem (harder than NP), there is very unlikely to a polynomial-time algorithm. 

Suppose you have an $n \times n$ matrix $M$ that you want to compute the permanent of. Construct a graph on $2n+1$ vertices, where there is a special vertex $v$ which has one copy, and $n$ of the vertices $x_1$ $\ldots$, $x_n$ have only one copy, while the remaining $n$ vertices $y_1$, $\ldots$, $y_n$, have $n$ copies each. On the $i$'th copy of vertex $y_j$, connect it to $x_i$. Connect the $i$'th copy of $y_j$ to $v$ if and only if the entry $M(i,j) = 1$. 

Now, for $x_i$ to be in the spanning tree, you must have chosen the $i$'th copy of $y_j$ for some $j$. So we must have a permutation $\pi$, where we have chosen the $\pi(j)$ copy of vertex $y_j$. For this permutation to be a spanning tree we must have $M(\pi(j), j) = 1$ for each vertex $j$. This shows that the permutation $\pi$ yields a spanning tree if and only if it contributes $1$ to the permanent of $M$. Thus, the number of spanning trees is the permanent of $M$.