We introduce a mathematical model to treat the polymerase chain reaction (PCR), where we regard the accumulation of new molecules during a PCR cycle as a randomly bifurcating tree. This model enables us to compute an approximate formula for the distribution of the number of replications that have occurred between a pair of molecules, which depends on the efficiency lambda of the reaction, the number N0 of template molecules at the beginning of the PCR and the number c of PCR cycles. The reliability of the approximation is tested by computer simulations. Finally, to model the effect of the intrinsic error rate of the polymerase, we superimpose a substitution process on the tree. The resulting closed formula for the distribution of pairwise differences of sequences as a function of error rate mu and efficiency lambda can be used to estimate the error rate, if lambda is known.