The distribution of interior branch lengths in phylogenetic trees under constant-rate birth-death models

Phylogenetic trees represent the evolutionary relationships between extant lineages, omitting extinct or non-sampled lineages. Extending the work of Stadler and collaborators, this paper focuses on the distribution of branch lengths in phylogenetic trees arising under a constant-rate birth-death model. We derive branch length distributions of interior branches with and without random sampling of individuals of the extant population. We establish that branches connected to the tree leaves and interior branches behave very differently under sampling: pendant branches get longer as the sampling probability reaches zero, whereas the distribution of interior branches quickly reaches an asymptotic state.