The Critical Beta-splitting Random Tree III: The exchangeable partition representation and the fringe tree

In the critical beta-splitting model of a random $n$-leaf rooted tree, clades are recursively split into sub-clades, and a clade of $m$ leaves is split into sub-clades containing $i$ and $m-i$ leaves with probabilities $\propto 1/(i(m-i))$. Study of structure theory and explicit quantitative aspects of the model is an active research topic. It turns out that many results have several different proofs, and detailed studies of analytic proofs are given elsdewhere (via analysis of recursions and via Mellin transforms). This article describes two core probabilistic methods for studying $n \to \infty$ asymptotics of the basic finite-$n$-leaf models. (i) There is a canonical embedding into a continuous-time model, that is a random tree CTCS(n) on $n$ leaves with real-valued edge lengths, and this model turns out to be more convenient to study. The family (CTCS(n), $n \ge 2)$ is consistent under a ``delete random leaf and prune" operation. That leads to an explicit inductive construction (the {\em growth algorithm}) of (CTCS(n), $n \ge 2)$ as $n$ increases, and then to a limit structure CTCS$(\infty)$ which can be formalized via exchangeable partitions, in some ways analogous to the Brownian continuum random tree. (ii) There is an explicit description of the limit fringe distribution relative to a random leaf, whose graphical representation is essentially the format of the cladogram representation of biological phylogenies.