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A popular line of research in evolutionary biology is the use oftime-calibrated phylogenies for the inference of diversification processes.This requires computing the likelihood of a given ultrametric tree as thereconstructed tree produced by a given model of diversification. Etienne &Rosindell (2012) proposed a lineage-based model of diversification, calledprotracted speciation, where species remain incipient during a random durationbefore turning good species, and showed that this can explain the slowdown inlineage accumulation observed in real phylogenies. However, they were unable toprovide a general likelihood formula. Here, we present a likelihood formula forprotracted speciation models, where rates at which species turn good or becomeextinct can depend both on their age and on time. Our only restrictiveassumption is that speciation rate does not depend on species status.  Our likelihood formula utilizes a new technique, based on the contour of thephylogenetic tree and first developed in Lambert (2010). We consider thereconstructed trees spanned by all extant species, by all good extant species,or by all representative species, which are either good extant species orincipient species representative of some good extinct species. Specifically, weprove that each of these trees is a coalescent point process, that is, aplanar, ultrametric tree where the coalescence times between two consecutivetips are independent, identically distributed random variables. We characterizethe common distribution of these coalescence times in some, biologicallymeaningful, special cases for which the likelihood reduces to an elegantanalytical formula or becomes numerically tractable.

The reconstructed tree in the lineage-based model of protracted speciation