On the Genealogy of Populations: Trees, Branches and Offsprings

We consider a neutral haploid population whose generations are not overlapping and whose size is large and constantly of $N$ individuals. Any generation is replaced by a new one and any individual has a single parent. We do not choose the stochastic rule which assigns the number of offsprings to any individual since results do not depend on the details of the dynamics, and, as a consequence, the model is parameter free. The genealogical tree is very complex, and distances between individuals (number of generations from the common ancestor) are distributed according to probability density which remains random in the thermodynamic limit (large population). We give a theoretical and numerical description of this distribution and we also consider the dynamical aspects of the problem describing the time evolution of the maximum and mean distances in a single population.