  The relative roles of mergers and star formation in regulating galaxy growth
are still a matter of intense debate. We here present our DECODE, a new
Discrete statistical sEmi-empiriCal mODEl specifically designed to predict
rapidly and efficiently, in a full cosmological context, galaxy assembly and
merger histories for any given input stellar mass-halo mass (SMHM) relation.
DECODE generates object-by-object dark matter merger trees (hence discrete)
from accurate subhalo mass and infall redshift probability functions (hence
statistical) for all subhaloes, including those residing within other
subhaloes, with virtually no resolution limits on mass or volume. Merger trees
are then converted into galaxy assembly histories via an input, redshift
dependent SMHM relation, which is highly sensitive to the significant
systematics in the galaxy stellar mass function and on its evolution with
cosmic time. DECODE can accurately reproduce the predicted mean galaxy merger
rates and assembly histories of hydrodynamic simulations and semi-analytic
models, when adopting in input their SMHM relations. In the present work we use
DECODE to prove that only SMHM relations implied by stellar mass functions
characterized by large abundances of massive galaxies and significant redshift
evolution, at least at $M_\star \gtrsim 10^{11} \, M_\odot$, can simultaneously
reproduce the local abundances of satellite galaxies, the galaxy (major merger)
pairs since $z \sim 3$, and the growth of Brightest Cluster Galaxies. The same
models can also reproduce the local fraction of elliptical galaxies, on the
assumption that these are strictly formed by major mergers, but not the full
bulge-to-disc ratio distributions, which require additional processes.
