  We extend the classical one-parameter Yule-Simon law to a version depending
on two parameters, which in part appeared in Bertoin [2019] in the context of a
preferential attachment algorithm with fading memory. By making the link to a
general branching process with age-dependent reproduction rate, we study the
tail-asymptotic behavior of the two-parameter Yule-Simon law, as it was already
initiated in the mentioned paper. Finally, by superposing mutations to the
branching process, we propose a model which leads to the full two-parameter
range of the Yule-Simon law, generalizing thereby the work of Simon [1955] on
limiting word frequencies.
