Take time-calibrated phylogenies as data.
For example, topological balance beta, and relative branch lengths gamma.
Lineage-based model– birth-death model. Birth and death rates may depend on time t, number n of standing particles, a non-heritable trait a, and a heritable trait i. Assume mother keeps the trait, and the daughter splits to the right. Remove extinct tips.
Lambert & Stadler, 2013: - same topology as Yule trees iff birth rate in terms of t and n only, and d depends on t, n, and a. -> such trees cannot explain imbalance - likelihood of reconstructed trees always has an explicit product form iff b is a function of t and d is a function of t and a only.
Coalescent point process– oriented tree whose node depths form a sequence of iid random variables $H_i$ killed at its first value larger than T.
If b is a function of t and d is a function of t and a only, the reconstructed tree is a CPP with typical node depth H, whose law is implicitly defined. Also works with extinction rate.
Application to bird phylogeny– extinction rate increase with age.
Now particles are populations. Every population starts as an incipient population, then they become “good” at some random time to make a new species.
Reconstructed tree spanned by extant representative populations at T is a coalescent point process with a node depth with a computable density.