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 ⋃G_n=G. Pemantle’s arguments imply that the uniform measures on spanning trees of G_n converge weakly to an Aut (G)-invariant measure μ_G on essential spanning forests of G. We show that if G is a connected, amenable graph and Γ⊂Aut (G) acts quasitransitively on G, then μ_G is the unique Γ-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 Γ≅ℤ^d. Lyons discovered the error and asked about the more general statement that we prove.