  We analyze the eigenvalues of the adjacency matrices of a wide variety of
random trees. Using general, broadly applicable arguments based on the
interlacing inequalities for the eigenvalues of a principal submatrix of a
Hermitian matrix and a suitable notion of local weak convergence for an
ensemble of random trees, we show that the empirical spectral distributions for
each of a number of random tree models converge to a deterministic (model
dependent) limit as the number of vertices goes to infinity.
  We conclude for ensembles such as the linear preferential attachment models,
random recursive trees, and the uniform random trees that the limiting spectral
distribution has a set of atoms that is dense in the real line. We obtain
precise asymptotics on the mass assigned to zero by the empirical spectral
measures via the connection with the cardinality of a maximal matching.
Moreover, we show that the total weight of a weighted matching is
asymptotically equivalent to a constant multiple of the number of vertices when
the edge weights are independent, identically distributed, non-negative random
variables with finite expected value.
  We greatly extend a celebrated result obtained by Schwenk for the uniform
random trees by showing that, under mild conditions, with probability
converging to one, the spectrum of a realization is shared by at least one
other tree.
  For the the linear preferential attachment model with parameter $a > -1$, we
show that the suitably rescaled $k$ largest eigenvalues converge jointly.
