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

In the critical beta-splitting model of a random $n$-leaf rooted tree, clades\nare recursively split into sub-clades, and a clade of $m$ leaves is split into\nsub-clades containing $i$ and $m-i$ leaves with probabilities $\\propto\n1/(i(m-i))$. Study of structure theory and explicit quantitative aspects of the\nmodel is an active research topic. It turns out that many results have several\ndifferent proofs, and detailed studies of analytic proofs are given elsdewhere\n(via analysis of recursions and via Mellin transforms). This article describes\ntwo core probabilistic methods for studying $n \\to \\infty$ asymptotics of the\nbasic finite-$n$-leaf models.\n (i) There is a canonical embedding into a continuous-time model, that is a\nrandom tree CTCS(n) on $n$ leaves with real-valued edge lengths, and this model\nturns out to be more convenient to study. The family (CTCS(n), $n \\ge 2)$ is\nconsistent under a ``delete random leaf and prune" operation. That leads to an\nexplicit inductive construction (the {\\em growth algorithm}) of (CTCS(n), $n\n\\ge 2)$ as $n$ increases, and then to a limit structure CTCS$(\\infty)$ which\ncan be formalized via exchangeable partitions, in some ways analogous to the\nBrownian continuum random tree.\n (ii) There is an explicit description of the limit fringe distribution\nrelative to a random leaf, whose graphical representation is essentially the\nformat of the cladogram representation of biological phylogenies.\n