Gaussian approximations for phylogenetic branch length statistics under stochastic models of biodiversity

Stein's method for Gaussian approximations and derived results are used to study the distribution of two phylogenetic branch length statistics: the total height of cherries and the sum of external branch lengths. The Gaussian approximations are obtained under a particular model of phylogenetic tree recently introduced by Popovic. Under an appropriate normalization the model is shown to behave similarly as the coalescent, and the approximations given here are also valid in this context.