  Deterministically growing (wild-type) populations which seed stochastically
developing mutant clones have found an expanding number of applications from
microbial populations to cancer. The special case of exponential wild-type
population growth, usually termed the Luria-Delbr\"uck or Lea-Coulson model, is
often assumed but seldom realistic. In this article we generalise this model to
different types of wild-type population growth, with mutants evolving as a
birth-death branching process. Our focus is on the size distribution of clones
- that is the number of progeny of a founder mutant - which can be mapped to
the total number of mutants. Exact expressions are derived for exponential,
power-law and logistic population growth. Additionally for a large class of
population growth we prove that the long time limit of the clone size
distribution has a general two-parameter form, whose tail decays as a
power-law. Considering metastases in cancer as the mutant clones, upon
analysing a data-set of their size distribution, we indeed find that a
power-law tail is more likely than an exponential one.
