  We study the evolution of stellar mass in galaxies as a function of host halo
mass, using the "MPA" and "Durham" semi-analytic models, implemented on the
Millennium Run simulation. The results from both models are similar. We find
that about 45% of the stellar mass in central galaxies in present-day halos
less massive than ~10^{12} Msun/h is already in place at z~1. This fraction
increases to ~65% for more massive halos. The peak of star formation efficiency
shifts toward lower mass halos from z~1 to z~0. The stellar mass in low-mass
halos grows mostly by star formation since z~1, while in high-mass halos most
of the stellar mass is assembled by mergers. These trends are clear indications
of "halo downsizing". We compare our findings to the results of the
phenomenological method developed by Zheng, Coil & Zehavi (2007). The
theoretical predictions are in qualitative agreement with these results,
however there are large discrepancies. The most significant one concerns the
amount of stars already in place in the progenitor galaxies at z~1, which is
about a factor of two larger in both semi-analytic models. We also use the
semi-analytic catalogs to test different assumptions made in that work, and
illustrate the importance of smooth accretion of dark matter when estimating
the mergers contribution. We demonstrate that methods studying galaxy evolution
from the galaxy-halo connection are powerful in constraining theoretical models
and can guide future efforts of modeling galaxy evolution. Conversely,
semi-analytic models serve an important role in improving such methods.
