The patterns obtained are complex, and at first sight it may seem surprising that such complexity arises from such a simple algorithm, but a qualitative understanding is straightforward. A single-cell bump on a straight edge is amplified because it has three growth sites (unoccupied neighbors) while each cell along the edge has only one growth site, and so the bump is more likely to capture a wandering particle.
That is, initial fluctuations from the straight-line edge are likely to grow into larger branches. By the same process, bumps along a branch lead to the formation of smaller branches off the larger, suggesting that DLA clusters have fractal properties.