  We study a simple model of DNA evolution in a growing population of cells.
Each cell contains a nucleotide sequence which randomly mutates at cell
division. Cells divide according to a branching process. Following typical
parameter values in bacteria and cancer cell populations, we take the mutation
rate to zero and the final number of cells to infinity. We prove that almost
every site (entry of the nucleotide sequence) is mutated in only a finite
number of cells, and these numbers are independent across sites. However
independence breaks down for the rare sites which are mutated in a positive
fraction of the population. The model is free from the popular but disputed
infinite sites assumption. Violations of the infinite sites assumption are
widespread while their impact on mutation frequencies is negligible at the
scale of population fractions. Some results are generalised to allow for cell
death, selection, and site-specific mutation rates. For illustration we
estimate mutation rates in a lung adenocarcinoma.
