  We use the galaxy stellar mass and halo merger tree information from the
semi-analytic model galaxy catalogue of Font et al. (2009) to examine the
accretion of galaxies into a large sample of groups and clusters, covering a
wide range in halo mass (10E12.9 to 10E15.3 Msun/h), and selected from each of
four redshift epochs (z=0, 0.5, 1.0 and 1.5). We find that clusters at all
examined redshifts have accreted a significant fraction of their final galaxy
populations through galaxy groups. A 10E14.5 Msun/h mass cluster at z=0 has, on
average, accreted ~ 40% of its galaxies (Mstellar > 10E9 Msun/h) from halos
with masses greater than 10E13 Msun/h. Further, the galaxies which are accreted
through groups are more massive, on average, than galaxies accreted through
smaller halos or from the field population. We find that at a given epoch, the
fraction of galaxies accreted from isolated environments is independent of the
final cluster or group mass. In contrast, we find that observing a cluster of
the same halo mass at each redshift epoch implies different accretion rates of
isolated galaxies, from 5-6 % per Gyr at z=0 to 15% per Gyr at z=1.5. We find
that combining the existence of a Butcher Oemler effect at z=0.5 and the
observations that galaxies within groups display significant environmental
effects with galaxy accretion histories justifies striking conclusions. Namely,
that the dominant environmental process must begin to occur in halos of 10E12
-- 10E13 Msun/h, and act over timescales of > 2 Gyrs. This argues in favor of a
mechanism like "strangulation", in which the hot halo of a galaxy is stripped
upon infalling into a more massive halo . This simple model predicts that by
z=1.5 galaxy groups and clusters will display little to no environmental
effects.
