Suppose that we sample the leaves of edge-weighted tree with n leaves in a uniform random order and record the lengths of the subtrees spanned by the first k leaves (that is, in biological terms, the phylogenetic diversity of the first k taxa) for k between 2 and n. In joint work with Daniel Lanoue (UC Berkeley) we consider the question, ``Can we reconstruct the tree (up to isomorphism) from the joint probability distribution of this random increasing sequence of lengths?" The answer is affirmative if we know a priori that the tree belongs to one of a number of families, but the general question is still open.
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