Statistical properties of genealogical trees

We analyse the statistical properties of genealogical trees in a neutral model of a closed population with sexual reproduction and non-overlapping generations. By reconstructing the genealogy of an individual from the population evolution, we measure the distribution of ancestors appearing more than once in a given tree. After a transient time, the probability of repetition follows, up to a rescaling, a stationary distribution which we calculate both numerically and analytically. This distribution exhibits a universal shape with a non-trivial power law which can be understood by an exact, though simple, renormalization calculation. Some real data on human genealogy illustrate the problem, which is relevant to the study of the real degree of diversity in closed interbreeding communities.