  We discuss a notion of convergence for binary trees that is based on subtree
sizes. In analogy to recent developments in the theory of graphs, posets and
permutations we investigate some general aspects of the topology, such as a
characterization of the set of possible limits and its structure as a metric
space. For random trees the subtree size topology arises in the context of
algorithms for searching and sorting when applied to random input, resulting in
a sequence of nested trees. For these we obtain a structural result based on a
local version of exchangeability. This in turn leads to a central limit
theorem, with possibly mixed asymptotic normality.
