  We discuss the spectral asymptotics of some open subsets of the real line
with random fractal boundary and of a random fractal, the continuum random
tree. In the case of open subsets with random fractal boundary we establish the
existence of the second order term in the asymptotics almost surely and then
determine when there will be a central limit theorem which captures the
fluctuations around this limit. We will show examples from a class of random
fractals generated from Dirichlet distributions as this is a relatively simple
setting in which there are sets where there will and will not be a central
limit theorem. The Brownian continuum random tree can also be viewed as a
random fractal generated by a Dirichlet distribution. The first order term in
the spectral asymptotics is known almost surely and here we show that there is
a central limit theorem describing the fluctuations about this, though the
positivity of the variance arising in the central limit theorem is left open.
In both cases these fractals can be described through a general
Crump-Mode-Jagers branching process and we exploit this connection to establish
our central limit theorems for the higher order terms in the spectral
asymptotics. Our main tool is a central limit theorem for such general
branching processes which we prove under conditions which are weaker than those
previously known.
