A Monte Carlo model is used to simulate the morphological evolution of crystals growing from a vapor phase. The model combines nutrient diffusion, based on a modified diffusion-limited aggregation (DLA) process, with anisotropic surface-attachment kinetics and surface diffusion. First- and second-nearest-neighbor interactions are taken into account. Through a systematic variation of the simulation (growth) parameters, the whole range of growth morphologies observed in nature is recovered. Depending on the imposed temperature, supersaturation, and bond energies, the model crystals are found to grow either fully faceted or with (dendritic) protrusions which, in turn, can either be branchless or develop side branches with smooth or split tips. The successive transitions from compact faceted to open dendritic morphologies are seen to arise as a collective result of nutrient diffusion and interface kinetics. The scale invariance exhibited by DLA does not generally hold in (dendritic) crystal growth. Thus, this work is not only an extension of previous DLA-type simulations, but affords physical insight into the nature of the interaction of crystal morphology with surface and volume processes. © 1988 The American Physical Society.