I asked Daniel Spielman a related question, which was what the most conceptual proof of the matrix-tree theorem is, without appealing to the real numbers. He suggested these notes by Nikhil Srivastava.

Matrices are used, but really the linear algebra looks to me like it can be turned into combinatorics on graphs in a straightforward manner, so his answer constitutes a nice answer to this question as well. In any event, it's a nice short conceptual proof of the matrix-tree theorem which I hadn't known about.

I asked Daniel Spielman a related question, which was what the most conceptual proof of the matrix-tree theorem is, without appealing to the real numbers. He suggested these notes by Nikhil Srivastava.

Matrices are used, but really the linear algebra looks to me like it can be turned into combinatorics on graphs in a straightforward manner, so his answer constitutes a nice answer to this question as well. In any event, it's a nice short conceptual proof of the matrix-tree theorem which I hadn't known about.