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 $\lambda$ for time $t$. The length of a randomly chosen edge (either interior or pendant) in the resulting tree has an expected value that rapidly converges to $\frac{1}{2\lambda}$ as $t$ grows and thus is essentially independent of $t$. However, the behavior of the length $L$ of the longest pendant edge reveals remarkably different behavior: $L$ converges to $t/2$ as the expected number of leaves grows. Extending this model to allow an extinction rate $\mu$ (where $\mu<\lambda$), we also establish a similar result for birth-death trees, except that $t/2$ is replaced by $t/2 \cdot (1-\mu/\lambda)$. This "complete" tree may contain subtrees that have died out before time $t$; for the "reduced tree" that just involves the leaves present at time $t$ and their direct ancestors, the longest pendant edge length $L$ again converges to $t/2$. 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.].