Colored Genealogical Trees and Coalescent Theory

We introduce a colored coalescent process which recovers random colored genealogical trees. Here a colored genealogical tree has its vertices colored black or white. Moving backward along the colored genealogical tree, the color of vertices may change only when two vertice coalesce. The rule that governs the change of color involves a parameter x. When x = 1/2, the colored coalescent process can be derived from a variant of the Wright-Fisher model for a haploid population in population genetics. Explicit computations of the expectation and the cumulative distribution function of the coalescent time are carried out. For example, our calculation shows that when x = 1/2, for a sample of n colored individuals, the expected time for the colored coalescent process to reach a black MRAC or a white MRAC, respectively, is 3−2/n. On the other hand, the expected time for the colored coalescent process to reach a MRAC, either black or white, is 2 − 2/n, which is the same as that for the standard Kingman coalescent process. This colored coalescent process with a color mutation process superimposed is also studied in explicit details.