An abstract tropical curve is essentially a "metric graph" (with some extra data). There is a reasonable theory of "divisors" in this setting. For example, there is a tropical analogue of the Picard group. It turns out the combinatorics and geometry of the Picard group is intimately related to the classical Matrix-Tree Theorem. I will describe some of these connections. If time permits I will discuss a recent generalizations to (metric) regular matroids. No prior knowledge in the subject will be assumed. This talk is based on joint works with M. Baker, G. Kuperberg, A. Yang, and with Esco He, Avery St. Dizier.