  An essential spanning forest of an infinite graph $G$ is a spanning forest of
$G$ in which all trees have infinitely many vertices. Let $G_n$ be an
increasing sequence of finite connected subgraphs of $G$ for which $\bigcup
G_n=G$. Pemantle's arguments imply that the uniform measures on spanning trees
of $G_n$ converge weakly to an $\operatorname {Aut}(G)$-invariant measure
$\mu_G$ on essential spanning forests of $G$. We show that if $G$ is a
connected, amenable graph and $\Gamma \subset \operatorname {Aut}(G)$ acts
quasitransitively on $G$, then $\mu_G$ is the unique $\Gamma$-invariant measure
on essential spanning forests of $G$ for which the specific entropy is maximal.
This result originated with Burton and Pemantle, who gave a short but incorrect
proof in the case $\Gamma\cong\mathbb{Z}^d$. Lyons discovered the error and
asked about the more general statement that we prove.
