Before revision:  Assuming that the population grows from one to many cells and that mutation rates are small, we prove limit theorems. At a single site (position) on the genetic sequence, the number of mutated cellsfollows a Luria-Delbr\"uck distribution. Across multiple sites , the joint distribution of mutation frequencies reflects the structure of the evolutionary tree. Taking the number of sites to infinity, the site frequency spectrum is deterministic at small frequencies, transitioning to random at large frequencies. Heterogeneous mutation ratesand selection are discussed. For an example application we estimate mutation rates in a lung adenocarcinoma.
After revision:  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 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. We also estimate mutation rates in a lung adenocarcinoma.