  We study the fractal structure of Diffusion-Limited Aggregation (DLA)
clusters on the square lattice by extensive numerical simulations (with
clusters having up to $10^8$ particles). We observe that DLA clusters undergo
strongly anisotropic growth, with the maximal growth rate along the axes. The
naive scaling limit of a DLA cluster by its diameter is thus deterministic and
one-dimensional. At the same time, on all scales from the particle size to the
size of the entire cluster it has non-trivial box-counting fractal dimension
which corresponds to the overall growth rate which, in turn, is smaller than
the growth rate along the axes. This suggests that the fractal nature of the
lattice DLA should be understood in terms of fluctuations around
one-dimensional backbone of the cluster.
