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

A note on limits of sequences of binary trees