According to the paper Benjamini, Kesten, Peres, and Schramm - Geometry of the uniform spanning forest: transitions in dimensions 4, 8, 12 (Annals, 2004), the diameter of the component graph of the uniform spanning forests (USF) on $d$-dimensional transitive graph is $\lfloor(d-1)/4\rfloor$.

I wonder if the diameter of the component graph of the USF on the amenable transitive graph with superpolynomal growth is infinite. Actually, I think the answer is yes for every transitive graph with superpolynomial growth. A concrete example is that I think the component graph of WUSF on 3-regular tree has infinite diameter. But 3-regular tree is not amenable.