Extensions of Matrix-Tree Theorem Kirchhoff's theorem relies on the notion of the Laplacian matrix of a graph that is equal to the difference between the graph's degree matrix (a diagonal matrix with vertex degrees on the diagonals) and its adjacency matrix (a $(0,1)$-matrix with $1$'s at places corresponding to entries where the vertices are adjacent and $0$'s otherwise).

It is known that for a given connected graph $G$ with $n$ labeled vertices, let $λ_1,λ_2,...,λ_{n−1}$ be the non-zero eigenvalues of its Laplacian matrix. Then the number of spanning trees of $G$ is $t(G)=\frac{1}{n} \lambda_1\lambda_2\cdots\lambda_{n-1}\,$

Now I want to extend it to the steiner tree. A $T$-Steiner tree is a subgraph of $G$ that is a tree and that spans $T$. When $T=V(G)$, a $T$-Steiner tree is a spanning tree. My question is that what is the connection between general $T$-Steiner tree and the Laplacian matrix? So far I didn't find any known results about this extension. Your comments and discussion are welcome.
Let me give a side answer to your question. Consider the variant where you only care about edge-minimal spanning trees: for 2 terminals $s$ and $t$, the problem is equivalent to counting simple $s,t$-paths, which is $\# P$-complete; also, the parameterized version where you ask for paths of length $k$ is $\# W[1]$-hard. If you care about arbitrary $T$-steiner trees (i.e. not necessarily minimal), I expect that it is also $\# P$-hard though I'm not aware of any reference.
These hardness results suggest that it's unlikely to find some "nice formula" for the number of $T$-steiner trees.

The references are: "The complexity of enumeration and reliability problems" by L. G. Valiant, and "The parameterized complexity of counting problems" by J. Flum and M. Grohe.