We present the first analytical and numerical studies of the initial stage of the branching process based on an interface dynamics streamer model in the fully 3-D case. This model follows from fundamental considerations on charge production by impact ionization and balance laws, and leads to an equation for the evolution of the interface between ionized and non-ionized regions. We compare some experimental patterns with the numerically simulated ones, and give an explicit expression for the growth rate of harmonic modes associated with the perturbation of a symmetrically expanding discharge. By means of full numerical simulation, the splitting and formation of characteristic tree-like patterns of electric discharges is observed and described.