  We use the random self-similarity of the continuum random tree to show that
it is homeomorphic to a post-critically finite self-similar fractal equipped
with a random self-similar metric. As an application we determine the mean and
almost-sure leading order behaviour of the high frequency asymptotics of the
eigenvalue counting function associated with the natural Dirichlet form on the
continuum random tree. We also obtain short time asymptotics for the trace of
the heat semigroup and the annealed on-diagonal heat kernel associated with
this Dirichlet form.
