  We present a generalization of the Yule model for macroevolution in which,
for the appearance of genera, we consider point processes with the order
statistics property, while for the growth of species we use nonlinear
time-fractional pure birth processes or a critical birth-death process.
Further, in specific cases we derive the explicit form of the distribution of
the number of species of a genus chosen uniformly at random for each time.
Besides, we introduce a time-changed mixed Poisson process with the same
marginal distribution as that of the time-fractional Poisson process.
