  We consider a Yule process until the total population reaches size $n\gg 1$,
and assume that neutral mutations occur with high probability $1-p$ (in the
sense that each child is a new mutant with probability $1-p$, independently of
the other children), where $p=p_n\ll 1$. We establish a general strategy for
obtaining Poisson limit laws for the number of subpopulations exceeding a given
size and apply this to some mutation regimes of particular interest. Finally,
we give an application to subcritical Bernoulli bond percolation on random
recursive trees with percolation parameter $p_n$ tending to zero.
