According to the paper [Benjamini, Kesten, Peres, and Schramm - Geometry of the Uniform Spanning Forest: Transitions in Dimensions 4, 8, 12](https://arxiv.org/abs/math/0107140), we have that the diameter of the component graph of the 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.