New analytic results for speciation times in neutral models.

In this paper, we investigate the standard Yule model, and a recently studied model of speciation and extinction, the "critical branching process." We develop an analytic way-as opposed to the common simulation approach-for calculating the speciation times in a reconstructed phylogenetic tree. Simple expressions for the density and the moments of the speciation times are obtained. Methods for dating a speciation event become valuable, if for the reconstructed phylogenetic trees, no time scale is available. A missing time scale could be due to supertree methods, morphological data, or molecular data which violates the molecular clock. Our analytic approach is, in particular, useful for the model with extinction, since simulations of birth-death processes which are conditioned on obtaining n extant species today are quite delicate. Further, simulations are very time consuming for big n under both models.