Uniform spanning-tree limit of the random-cluster model

If $p,q\downarrow 0$ with $q/p\to 0$, the random-cluster model on a finite connected graph converges to the uniform spanning tree. Relative to a tree, an acyclic configuration with $k>1$ components pays a factor $(q/p{)}^{k-1}$, while every additional cycle pays a factor asymptotic to $p$.