We study the evaluation of interface morphologies in free dendritic growth with heat or mass transport by either diffusion alone or by both diffusion and convection. This is done by quantitative numerical simulations; in particular, we try to capture the effects of crystalline anisotropy accurately, avoiding artefacts of the numerical grid. Tracking the interface dynamically, we find oscillating structures that can be related to transient behaviour in extended systems. The set-up of our program allows us to directly compare the effects of diffusion on the evolving patterns versus those of convection. Our aim is to study how the morphology diagram suggested by Brener et al. (Phys. Rev. E 54 (1996) 2714) is to be extended as a function of parameters describing convention.