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The multispecies coalescent process models the genealogical relationships ofgenes sampled from several species, enabling useful predictions about phenomenasuch as the discordance between the gene tree and the species phylogeny due toincomplete lineage sorting. Conversely, knowledge of large collections of genetrees can inform us about several aspects of the species phylogeny, such as itstopology and ancestral population sizes. A fundamental open problem in thiscontext is how to efficiently compute the probability of a gene tree topology,given the species phylogeny. Although a number of algorithms for this task havebeen proposed, they either produce approximate results, or, when they areexact, they do not scale to large data sets. In this paper, we present someprogress towards exact and efficient computation of the probability of a genetree topology. We provide a new algorithm that, given a species tree and thenumber of genes sampled for each species, calculates the probability that thegene tree topology will be concordant with the species tree. Moreover, weprovide an algorithm that computes the probability of any specific gene treetopology concordant with the species tree. Both algorithms run in polynomialtime and have been implemented in Python. Experiments show that they are ableto analyse data sets where thousands of genes are sampled, in a matter ofminutes to hours.

Computing the probability of gene trees concordant with the species tree in the multispecies coalescent