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 (1925)'s 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 $A$ and size $S$ of any randomly chosen genus and that the pairs $(A_i,S_i)$ are equally distributed and independent across genera. This property extends to triples $(H_i,A_i,S_i)$, where $(H_i)$ are the coalescence times of the genus phylogeny, even when species diversification within genera follows any integer-valued process, including species extinctions. Studying $(A,S)$ in this broader context allows us to identify cases where $S$ has a power-law tail distribution, with new applications to urn schemes.

33 pages, 4 figures. To appear in theme issue of Philosophical Transactions of the Royal Society Series B on "Phylogenetic models a century beyond G. U. Yule's 'mathematical theory of evolution'"