On leaf related statistics in recursive tree models

Using a bijection between a uniformly random permutation and a uniform recursive tree (URT), we give a simple proof of a recent result of Zhang that shows the asymptotic normality of the number of leaves in a URT with convergence rates. We also show that a similar result holds for a more general class of statistics related to URTs, that we call the number of runs of leaves in a URT. The second, and the main, purpose of the current note is to introduce and to study a non-uniform recursive tree model by exploiting the bijection between permutations and recursive trees. This may provide a useful framework for constructing various types of random trees.