graphs $\gh$ [**tight**]{} if for each $R$, the collection of corresponding distributions $\nu_R(\gh)$ is tight. Note that any tight collection of finite graphs has a subsequence that possesses a random weak limit. Define the [**expected degree**]{} of a probability measure $\rtd$ on rooted graphs to be $$\expdeg (\rtd) := \int \deg_\gh(x) \,d\rtd(\gh, x) 
\,.$$ When the following integral converges, define the [**tree entropy**]{} of $\rtd$ to be $$\asym(\rtd)
:=
\int \Big(\log \deg_\gh(x) - \sum_{k \ge 1} {1 \over k} p_k(x;\gh)\Big)
\,d\rtd(\gh, x) 
\,.$$ Our terminology is justified by Theorems t.asym/ and t.amen-ent/ below. The integral converges, for example, when $\rtd$ has finite expected degree, i.e., $
\expdeg (\rtd) < \infty
$, by virtue of the inequality between the arithmetic and geometric means; see also \[c\].hbound/ and \[p\].positive/ below. If $\rtd$ is induced by a fixed transitive graph, $\gh$, of degree $d$, we write $$\asym(\gh) := \asym(\rtd)
=
\log d - \sum_{k \ge 1}
{1 \over k} p_k(o;\gh)$$ for the tree entropy of $\gh$, where $o$ is any vertex of $\gh$.

Our main theorem in this section is the following. It also suggests thinking of $\asym(\rtd)$ as a normalized logarithm of the determinant of the Laplacian. See \[t\].trlog/ for a more 