Statistical properties of genealogical trees

We analyze the statistical properties of genealogical trees in a neutral model of a closed population with sexual reproduction and nonoverlapping 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 nontrivial 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. [ S0031-9007(99) Modern man appeared on Earth some 10 5 years ago These repetitions are particularly apparent when we are faced with a small closed interbreeding population. Royal genealogy provides us with a nice example, since nobles usually married within their own castes. As an illustration to the problem, we have analyzed the repetitions in the genealogical tree of the English king Edward III (1312-1377) [5]. It contains almost 10 3 individuals, some of which appear more than once (and up to six times) in his tree. We show in We study here the statistics of repetitions in genealogical trees as a function of the population size and the generation in the past that we are looking at. The question that we are addressing can be put in the more general context of genetic diversity We have started by performing numerical simulations of a simple neutral model of a closed population evolving under sexual reproduction with nonoverlapping generations