Our second main aim is to give meaning to and to prove the formula $$\asym(\gh) = \log \Det \, \Delta_\gh$$ directly in terms of a normalized determinant of the graph Laplacian for infinite graphs. We shall also use this formula to prove inequalities for tree entropy and to calculate easily and quickly the classical tree entropy for Euclidean lattices. Indeed, this result, \[t\].trlog/, provides a quick way to derive virtually all the prior asymptotics of this type in the literature, while \[t\].asym/ gives the rest (and more).

We have alluded to the fact that tree entropy also arises as the entropy per vertex of a measure that is the weak limit of the uniform measures on spanning trees of finite graphs. To state this precisely, we must first recall the work of \[b\].Pemantle:ust/. He proved a conjecture of the present author, namely, that if an infinite connected graph $\gh$ is exhausted by a sequence of finite connected subgraphs $\gh_n$, then the weak limit of the uniform spanning tree measures on $\gh_n$ exists. However, it may happen that the limit measure is not supported on trees, but on forests. This limit measure is now called the [**free uniform spanning forest**]{} on 