The theory of quantum chromodynamics (QCD) predicts that at high energies, such as those investigated in deep inelastic scattering experiments, hadrons evolve into dense gluonic states described by the BFKL equation, and at very high densities, the more general BK equation. In certain approximations, the BK equation reduces to a well studied reaction-diffusion type nonlinear partial differential equation, the FKPP equation, for which analytical results are known. In this work, we model the BK equation using a classical branching process rooted in the dipole model of QCD evolution. Because the BK equation is inherently two dimensional, our model allows dipole impact parameters to occupy the full transverse space. A one dimensional limit of this model is studied as well. Results are compared with the predictions of the FKPP equation, and correlations between evolution at different impact parameters are presented. The general features of previously studied one dimensional impact parameter models are verified, but the details are refined in what we believe to be a more accurate model. / text