Abstract
Dating the tree of life is a task far more complicated that only determining the evolutionary relationships between species. It is therefore of interest to develop approaches able to deal with undated phylogenetic trees.The main result of this work is a method to compute probabilities of undated phylogenetic trees under piecewise-constant-birth-death-sampling models by constraining some of the divergence times to belong to given time intervals and by allowing diversification shifts on certain clades. The computation is quite fast since its time complexity is quadratic with the size of the tree topology and linear with the number of time constraints and of ''pieces'' in the model.The interest of this computation method is illustrated with three applications, namely, - to compute the exact distribution of the divergence times of a tree topology with temporal constraints, - to directly sample the divergence times of a tree topology, and - to test for a diversification shift at a given clade.