On the genealogy of a population of biparental individuals

If one goes backward in time, the number of ancestors of an individual doub les at each generation. This exponential growth very quickly exceeds the po pulation size, when this size is finite. As a consequence, the ancestors of a given individual cannot be all different and most remote ancestors are r epeated many times in any genealogical tree. The statistical properties of these repetitions in genealogical trees of individuals for a panmictic clos ed population of constant size N can be calculated. We show that the distri bution of the repetitions of ancestors reaches a stationary shape after a s mall number G(c) proportional to log N of generations in the past, that onl y about 80% of the ancestral population belongs to the tree (due to coalesc ence of branches), and that two trees for individuals in the same populatio n become identical after G(c) generations have elapsed. Our analysis is eas y to extend to the case of exponentially growing population. (C) 2000 Acade mic Press.