$\gh$, denoted $\fsf$, or $\fsf_\gh$ when we want to indicate the graph $\gh$. If $\gh$ is itself a tree, then this measure is trivial, namely, it is concentrated on $\{\gh\}$. Therefore, \[b\].Hag:umesft/ introduced another limit that had been considered on $\Z^d$ more implicitly by \[b\].Pemantle:ust/ and explicitly by \[b\].Hag:rcust/, namely, the weak limit of the uniform spanning tree measures on $\gh_n^*$, where $\gh_n^*$ is the graph $\gh_n$ with its boundary identified to a single vertex. As \[b\].Pemantle:ust/ showed, this limit also always exists on any graph and is now called the [**wired uniform spanning forest**]{}, denoted $\wsf$ or $\wsf_\gh$. It is clear that both $\fsf$ and $\wsf$ are concentrated on the set of spanning forests of $\gh$ that are [**essential**]{}, meaning that all their trees are infinite. Furthermore, since the limits exist regardless of the exhaustion chosen, both $\fsf_\gh$ and $\wsf_\gh$ are invariant under any automorphisms that $\gh$ may have. As shown by \[b\].Hag:rcust/, $\fsf_\gh = \wsf_\gh$ on every amenable transitive graph $\gh$ such as $\Z^d$. In fact, the proof of this result given by , is easily modified to show that the same holds for every quasi-transitive amenable graph, using the notion of natural frequency distribution recalled below in 