I am surprised no one mentioned the work of Alan Sokal and coworkers on precisely this issue
of weighted enumeration of spanning forests which is related to the $q\rightarrow 0$ limit of the
Potts model as well as the multivariate Tutte polynomial.

A determinant expression corresponds to a Fermionic (Grassmann/Berezin) integral with a quadratic
`action' in the exponential. There is an analogue of the matrix-tree theorem for spanning forests
with a Fermionic integral with quartic action see:
[Fermionic field theory for trees and forests](http://arxiv.org/abs/cond-mat/0403271)
which appeared in PRL.
Follow ups such as
[Grassmann Integral Representation for Spanning Hyperforests](http://arxiv.org/abs/0706.1509)
can be found by looking at 
[Sokal's papers on arxiv](http://arxiv.org/find/grp_math/1/au:+sokal/0/1/0/all/0/1).

Note that there was a whole semester at the Isaac Newton Institute revolving around this
topic: [Combinatorial identities and their applications in statistical mechanics](https://www.newton.ac.uk/event/csmw03/).
One can even watch the videos of the talks.
The one perhaps most relevant to this question is the talk by Andrea Sportiello in the fourth
workshop.