Figure 2

Representative symmetrical (a) and asymmetrical (b) genealogies for four sampled sequences. In all, there are six symmetrical and 12 asymmetrical labeled genealogies for these four sequences [8, 45]. This figure illustrates how the shape of a genealogy affects whether a mutation will lead to a singleton or shared polymorphism. This heterogeneity contributes to the variance of θ for the three new methods. This contribution is in addition to the heterogeneity of the genealogical branch lengths and Poisson mutation process [4, 5]. The diamond highlights the common ancestor of the (n - 1) basal group of the asymmetrical genealogy. The dotted and thin solid lines mark the basal branch leading to this ancestor and the external branches, respectively, where a mutation will result in a singleton. The thick solid lines denote the other internal branches where a mutation will lead to a shared polymorphism. Expected branch lengths are given in units of scaled coalescent time next to each internode (those of the symmetrical genealogy are specific for its particular labeled history). Although both genealogies have an expected overall length of 11/3, the total length of the internal branches where a shared polymorphism can arise is 7/3 for (a) but only 1/3 for (b).