Phylogenies of extant species are consistent with an infinite array of diversification histories
Time-calibrated molecular phylogenies of extant species (\"extant timetrees\") are widely used for estimating the dynamics of diversification rates (1-6) and testing for associations between these rates and environmental factors (5, 7) or species traits (8). However, there has been considerable debate surrounding the reliability of these inferences in the absence of fossil data (9-13), and to date this critical question remains unresolved. Here we mathematically clarify the precise information that can be extracted from extant timetrees under the generalized birth-death model, which underlies the majority of existing estimation methods. We prove that for a given extant timetree and a candidate diversification scenario, there exists an infinite number of alternative diversification scenarios that are equally likely to have generated a given tree. These \"congruent\" scenarios cannot possibly be distinguished using extant timetrees alone, even in the presence of infinite data. Importantly, congruent diversification scenarios can exhibit markedly different and yet plausible diversification dynamics, suggesting that many previous studies may have over-interpreted phylogenetic evidence. We show that sets of congruent models can be uniquely described using composite variables, which contain all available information about past dynamics of diversification (14); this suggests an alternative paradigm for learning about the past from extant timetrees.