  Galaxies grow primarily via accretion-driven star formation in discs and
merger-driven growth of bulges. These processes are implicit in semi-analytical
models of galaxy formation, with bulge growth in particular relating directly
to the hierarchical build-up of halos and their galaxies. In this paper, we
consider several implementations of two semi-analytical models. Focusing on
implementations in which bulges are formed during mergers only, we examine the
fractions of elliptical galaxies and both passive and star-forming disk
galaxies as functions of stellar and halo mass, for central and satellite
systems. This is compared to an observational cross-matched SDSS+RC3 z ~ 0
sample of galaxies with accurate visual morphological classifications and
M_{stellar} > 10^10.5 M_{sol}. The models qualitatively reproduce the observed
increase of elliptical fraction with stellar mass, and with halo mass for
central galaxies, supporting the idea that observed ellipticals form during
major mergers. However, the overall elliptical fraction produced by the models
is much too high compared with the z ~ 0 data. Since the "passive" -- i.e.
non-star-forming -- fractions are approximately reproduced, and since the
fraction which are star-forming disc galaxies is also reproduced, the problem
is that the models overproduce ellipticals at the expense of passive S0 and
spiral galaxies. Bulge-growth implementations (tuned to reproduce simulations)
which allow the survival of residual discs in major mergers still destroy too
much of the disc. Increasing the lifetime of satellites, or allowing
significant disc regrowth around merger remnants, merely increases the fraction
of star-forming disc galaxies. Instead, it seems necessary to reduce the mass
ratios of merging galaxies, so that most mergers produce modest bulge growth in
disc-galaxy remnants instead of ellipticals. [Abridged]
