  First, we revisit the stochastic Luria-Delbr\"uck model: a classic two-type
branching process which describes cell proliferation and mutation. We prove
limit theorems and exact results for the mutation times, clone sizes, and
number of mutants. Second, we extend the framework to consider mutations at
multiple sites along the genome. The number of mutants in the two-type model
characterises the mean site frequency spectrum in the multiple-site model. Our
predictions are consistent with previously published cancer genomic data.
