The Critical Beta-splitting Random Tree: Heights and Related Results

In the critical beta-splitting model of a random $n$-leaf binary tree, leaf-sets are recursively split into subsets, and a set of $m$ leaves is split into subsets containing $i$ and $m-i$ leaves with probabilities proportional to $1/{i(m-i)}$. We study the continuous-time model in which the holding time before that split is exponential with rate $h_{m-1}$, the harmonic number. We (sharply) evaluate the first two moments of the time-height $D_n$ and of the edge-height $L_n$ of a uniform random leaf (that is, the length of the path from the root to the leaf), and prove the corresponding CLTs. We find the limiting value of the correlation between the heights of two random leaves of the same tree realization, and analyze the expected number of splits necessary for a set of $t$ leaves to partially or completely break away from each other. We give tail bounds for the time-height and the edge-height of the {\em tree}, that is the maximal leaf heights. We show that there is a limit distribution for the size of a uniform random subtree, and derive the asymptotics of the mean size. Our proofs are based on asymptotic analysis of the attendant (sum-type) recurrences. The essential idea is to replace such a recursive equality by a pair of recursive inequalities for which matching asymptotic solutions can be found, allowing one to bound, both ways, the elusive explicit solution of the recursive equality. This reliance on recursive inequalities necessitates usage of Laplace transforms rather than Fourier characteristic functions.