Ages, sizes and (trees within) trees of taxa and of urns, from Yule to today.

Abstract

The paper written in 1925 by G. Udny Yule that we celebrate in this special issue introduces several novelties and results that we recall in detail. First, we discuss Yule's (1925)main legacies over the past century, focusing on empirical frequency distributions with heavy tails and random tree models for phylogenies. We estimate the year when Yule's work was re-discovered by scientists interested in stochastic processes of population growth (1948) and the year from which it began to be cited (1951, Yule's death). We highlight overlooked aspects of Yule's work (e.g. the Yule process of Yule processes) and correct some common misattributions (e.g. the Yule tree). Second, we generalize Yule's results on the average frequency of genera of a given age and size (number of species). We show that his formula also applies to the age [Formula: see text] and size [Formula: see text] of any randomly chosen genus and that the pairs [Formula: see text] are equally distributed and independent across genera. This property extends to triples [Formula: see text], where [Formula: see text] are the coalescence times of the genus phylogeny, even when species diversification within genera follows any integer-valued process, including species extinctions. Studying [Formula: see text] in this broader context allows us to identify cases where [Formula: see text] has a power-law tail distribution, with new applications to urn schemes.This article is part of the theme issue '"A mathematical theory of evolution": phylogenetic models dating back 100 years'.