Computing tree size under dynamical models of diversification

A phylogenetic tree has three types of attributes: size, shape (topology), and branch density. Phylodynamic studies are often motivated by questions regarding the size of clades, nevertheless, nearly all of the inference methods only make use of the other two attributes. In this paper, we ask whether there is additional information if we consider tree size more explicitly in phylodynamic inference methods. To address this question, we first needed to be able to compute the expected tree size distribution under a specified phylodynamic model; perhaps surprisingly, there is not a general method for doing so - it is known what this is under a Yule or constant rate birth-death model but not for the more complicated scenarios researchers are often interested in. We present three different solutions to this problem: using (i) the deterministic limit; (ii) master equations; and (iii) an ensemble moment approximation. Using simulations, we evaluate the accuracy of these three approaches under a variety of scenarios and alternative measures of tree size (i.e., sampling through time or only at the present; sampling ancestors or not). We then use the most accurate measures for the situation, to investigate the added informational content of tree size. We find that for two critical phylodynamic questions - (i) is diversification diversity dependent? and, (ii) can we distinguish between alternative diversification scenarios? - knowing the expected tree size distribution under the specified scenario provides insights that could not be gleaned from considering the expected shape and branch density alone. The contribution of this paper is both a novel set of methods for computing tree size distributions and a path forward for richer phylodynamic inference into the evolutionary and epidemiological processes that shape lineage trees.