  Let $G$ be a finite tree. For any matching $M$ of $G$, let $U(M)$ be the set
of vertices uncovered by $M$. Let $\mathcal{M}_G$ be a uniform random maximum
size matching of $G$. In this paper, we analyze the structure of
$U(\mathcal{M}_G)$. We first show that $U(\mathcal{M}_G)$ is a determinantal
process. We also show that for most vertices of $G$, the process
$U(\mathcal{M}_G)$ in a small neighborhood of that vertex can be well
approximated based on a somewhat larger neighborhood of the same vertex. Then
we show that the normalized Shannon entropy of $U(\mathcal{M}_G)$ can be also
well approximated using the local structure of $G$. In other words, in the
realm of trees, the normalized Shannon entropy of $U(\mathcal{M}_G)$ -- that
is, the normalized logarithm of the number of maximum size matchings of $G$ --
is a Benjamini-Schramm continuous parameter.
  We show that $U(\mathcal{M}_G)$ is a determinantal process through
establishing a new connection between $U(\mathcal{M}_G)$ and the adjacency
matrix of $G$. This result sheds a new light on the well-known fact that on a
tree, the number of vertices uncovered by a maximum size matching is equal to
the nullity of the adjacency matrix.
  Some of the proofs are based on the well established method of introducing a
new perturbative parameter, which we call temperature, and then define the
positive temperature analogue of $\mathcal{M}_G$, the so called monomer-dimer
model, and let the temperature go to zero.
