  We prove an analogue of Kirchhoff's matrix tree theorem for computing the
volume of the tropical Prym variety for double covers of metric graphs. We
interpret the formula in terms of a semi-canonical decomposition of the
tropical Prym variety, via a careful study of the tropical Abel-Prym map. In
particular, we show that the map is harmonic, determine its degree at every
cell of the decomposition, and prove that its global degree is $2^{g-1}$. Along
the way, we use the Ihara zeta function to provide a new proof of the analogous
result for finite graphs. As a counterpart, the appendix by Sebastian
Casalaina-Martin shows that the degree of the algebraic Abel-Prym map is
$2^{g-1}$ as well.
