Critical beta-splitting, via contraction

The critical beta-splitting tree, introduced by Aldous, is a Markov branching phylogenetic tree. Aldous and Pittel recently proved, amongst other results, a central limit theorem for the height of a random leaf. We give an alternative proof, via contraction methods for random recursive structures. These methods were developed by Neininger and R\"{u}schendorf, motivated by Pittel's article "Normal convergence problem? Two moments and a recurrence may be the clues." Aldous and Pittel estimated the leading order terms in the first two moments. More recently, Aldous and Janson obtained an asymptotic expansion for the average height. We show that a central limit theorem follows, and bound the distance to normality. Our results also apply to the continuous version of the model, in which branching times are exponential.