  In the classical model of Diffusion Limited Aggregation (DLA), introduced by
Witten and Sander, the process begins with a single particle cluster placed at
the origin of a space, and then, one at a time, particles make a random walk
from infinity until they collide with, and stick to, the existing cluster.
  We consider an analogous version of this process on large but finite graphs
with a designated source and sink vertex. Initially the cluster of halted
particles contains a single particle at the sink vertex. Starting one at a time
from the source, each particle makes a random walk in the direction of the sink
vertex. The particle halts at the last unoccupied vertex before the walk enters
the cluster for the first time, thus increasing the size of the cluster. This
continues until the source vertex becomes occupied, at which point the process
ends. We study the DLA process on several classes of layered graphs, including
Cayley trees of branching factor at least two with a sink vertex attached to
the leaves. We determine the finish time of the process for a given class of
graphs and show that the subcomponent of the final cluster linking source to
sink is essentially a unique path.
