  We consider random graphs with a given degree sequence and show, under weak
technical conditions, asymptotic normality of the number of components
isomorphic to a given tree, first for the random multigraph given by the
configuration model and then, by a conditioning argument, for the simple
uniform random graph with the given degree sequence. Such conditioning is
standard for convergence in probability, but much less straightforward for
convergence in distribution as here. The proof uses the method of moments, and
is based on a new estimate of mixed cumulants in a case of weakly dependent
variables. The result on small components is applied to give a new proof of a
recent result by Barbour and R\"ollin on asymptotic normality of the size of
the giant component in the random multigraph; moreover, we extend this to the
random simple graph.
