The authors introduce a biased diffusion model of aggregation at a surface, which reduces to a ballistic model in one limit. They characterise the structure of the aggregates by a variety of properties and find that it is a strong function of the parameter governing the diffusion process. For thin films there is a crossover between a regime where there are several highly ramified pseudo-one-dimensional clusters and another regime where there is a single cluster which spans the lattice in both directions.