Fig. 5
The ratio \(\frac {n_{b}}{n_{s}}\) of the upper bound on the minimum number of gene trees required to obtain a bipartition cover with probability q (Eq. 14) to the corresponding number of simulated gene trees required to obtain a bipartition cover with probability q. The ratio is plotted as a function of q, for several values of the number of species k. a T min=0.2. b T min=0.5. c T min=1.0. The y-axis is plotted on a logarithmic scale. Irregular spacing of q values is a result of our simulation procedure, in which each q is determined from 104 simulations at a fixed n s in the set {1,2,3,5,10,20,50,100,200,500}. Note that for some large values of n s at a fixed T min, all 104 simulations produced a bipartition cover, meaning that \(\hat {Q}_{n_{s}}=q=1\). In these cases, n b computed from Eq. 14 is infinite and we do not plot \(\frac {n_{b}}{n_{s}}\)