Motivation: Diversification rates and patterns may be inferred from reconstructed phylogenies. Both the time-dependent and the diversity-dependent birth-death process can produce the same observed patterns of diversity over time. To develop and test new models describing the macro-evolutionary process of diversification, generic and fast algorithms to simulate under these models are necessary. Simulations are not only important for testing and developing models but play an influential role in

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... e assessment of model fit. Results: In the present article, I consider as the model a global timedependent birth-death process where each species has the same rates but rates may vary over time. For this model, I derive the likelihood of the speciation times from a reconstructed phylogenetic tree and show that each speciation event is independent and identically distributed. This fact can be used to simulate efficiently reconstructed phylogenetic trees when conditioning on the number of species, the time of the process or both. I show the usability of the simulation by approximating the posterior predictive distribution of a birth-death process with decreasing diversification rates applied on a published bird phylogeny (family Cettiidae). Availability: The methods described in this manuscript are implemented in the R package TESS, available from the repository CRAN