Predicting Long Pendant Edges in Model Phylogenies, with Applications to Biodiversity and Tree Inference

In the simplest phylogenetic diversification model (the pure-birth Yule process), lineages split independently at a constant rate [Formula: see text] for time [Formula: see text]. The length of a randomly chosen edge (either interior or pendant) in the resulting tree has an expected value that rapidly converges to [Formula: see text] as [Formula: see text] grows and thus is essentially independent of [Formula: see text]. However, the behavior of the length [Formula: see text] of the longest pendant edge reveals remarkably different behavior: [Formula: see text] converges to [Formula: see text] as the expected number of leaves grows. Extending this model to allow an extinction rate [Formula: see text] (where [Formula: see text]), we also establish a similar result for birth–death trees, except that [Formula: see text] is replaced by [Formula: see text]. This “complete” tree may contain subtrees that have died out before time [Formula: see text]; for the “reduced tree” that just involves the leaves present at time [Formula: see text] and their direct ancestors, the longest pendant edge length [Formula: see text] again converges to [Formula: see text]. Thus, there is likely to be at least one extant species whose associated pendant branch attaches to the tree approximately half-way back in time to the origin of the entire clade. We also briefly consider the length of the shortest edges. Our results are relevant to phylogenetic diversity indices in biodiversity conservation, and to quantifying the length of aligned sequences required to correctly infer a tree. We compare our theoretical results with simulations and with the branch lengths from a recent phylogenetic tree of all mammals. [Birth–death process; phylogenetic diversification models; phylogenetic diversity.]