We investigate a neutral model for speciation and extinction, the constant rate birth-death process. The process is conditioned to have n extant species today, we look at the tree distribution of the reconstructed trees-i.e. the trees without the extinct species. Whereas the tree shape distribution is well-known and actually the same as under the pure birth process, no analytic results for the speciation times were known. We provide the distribution for the speciation times and calculate the expectations analytically. This characterizes the reconstructed trees completely. We will show how the results can be used to date phylogenies.

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 approachfor 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.

The Yule model is a widely used speciation model in evolutionary biology. Despite its simplicity many aspects of the Yule model have not been explored mathematically. In this paper, we formalise two analytic approaches for obtaining probability densities of individual branch lengths of phylogenetic trees generated by the Yule model. These methods are flexible and permit various aspects of the trees produced by Yule models to be investigated. One of our methods is applicable to a broader class of evolutionary processes, namely the Bellman-Harris models. Our methods have many practical applications including biodiversity and conservation related problems. In this setting the methods can be used to characterise the expected rate of biodiversity loss for Yule trees, as well as the expected gain of including the phylogeny in conservation management. We briefly explore these applications.

The reconstruction of large phylogenetic trees from data that violates clocklike evolution (or as a supertree constructed from any m input trees) raises a difficult question for biologists–how can one assign relative dates to the vertices of the tree? In this paper we investigate this problem, assuming a uniform distribution on the order of the inner vertices of the tree (which includes, but is more general than, the popular Yule distribution on trees). We derive fast algorithms for computing the probability that (i) any given vertex in the tree was the j–th speciation event (for each j), and (ii) any one given vertex is earlier in the tree than a second given vertex. We show how the first algorithm can be used to calculate the expected length of any given interior edge in any given tree that has been generated under either a constant-rate speciation model, or the coalescent model.

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