Abstract:

Aldous (1993, 2023) has introduced the "critical beta-splitting random tree"
(an interesting special case of more general "beta-splitting trees").  This
tree is constructed from a set of $n$ leaves, which are randomly partitioned
into two subsets, with the probability of subsets of sizes $j$ and $n-j$
being proportional to $1/j(n-j)$; continue recursively in each subset with
at least two elements. The result defines a binary tree with $n$ leaves. The
beta-splitting trees were partly motivated by the study of evolutionary
trees (although they are not realistic biological models). I see them as purely
mathematical objects with interesting properties, and I
will talk about some of these, using a variety of methods including both
probabilistic (continuous-time Markov processes; exchangeable partitions)
and analytic (an interesting Mellin transform). (Based on forthcoming joint
work with David Aldous, and perhaps others.)