By Khan R.A.

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1). The fraction of two-way coalescences becomes high as the population size passes 100, which is the square of the number of lineages. We can also examine, for N = 10, 000, the fraction of two-way coalescences with diﬀerent numbers of lineages (Fig. 2). These patterns can be summarized by saying that most coalescences will be two-way if n2 < N . However it is not obvious that having a modest fraction of three- or four-way coalescences will invalidate inference methods that assume the coalescent, so the coalescent may be a good approximation even when this condition is violated.

Each tree is speciﬁed by a given sequence of pairs of lineages that coalesce, plus the times of these coalescences. With n lineages, the sequence of coalescence events is speciﬁed by choice of pairs of lineages to coalesce. The total number of possibilities is n−1 i=1 n−i+1 2 = n! (n − 1)! 14) These diﬀerent possibilities are called labelled histories—they are diﬀerent trees in which we distinguish between the order of interior nodes in time. They were deﬁned by Edwards [8]; the formula counting them is given in that paper.

In addition, they are not inferences of the parameters of the underlying models. As such, they are not maximum likelihood estimates, but rather maxmimum posterior probability estimates (in a Bayesian framework they have posterior probabilities just as do the parameters). The question arises: is reconstructing the exact history a trivial pursuit? The quantities which are needed in further analyses are usually the underlying parameter values rather than the exact times of particular events. However, the ages of mutations or the depths of particular coalescences can serve as indications of whether an allele is not neutral, or a population size not constant.

### A bivariate extension of Bleimann-Butzer-Hahn operator by Khan R.A.

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