  The study of galaxy morphology is fundamental to understand the physical
processes driving the structural evolution of galaxies. It has long been known
that dense environments host high fractions of early-type galaxies and low
fractions of late-type galaxies, indicating that the environment affects the
structural evolution of galaxies. In this paper we present an analysis of the
morphological composition of red sequence galaxies in a sample of 9 galaxy
clusters at $0.8<z<1.5$ drawn from the HAWK-I cluster survey (HCS), with the
aim of investigating the evolutionary paths of galaxies with different
morphologies. We classify galaxies according to their apparent bulge-to-total
light ratio and compare with red sequence galaxies from the lower-redshift
WINGS and EDisCS surveys. We find that, while the HCS red sequence is dominated
by elliptical galaxies at all luminosities and stellar masses, the WINGS red
sequence is dominated by elliptical galaxies only at its bright end
($M_V<-21.0$ mag), while S0s become the most frequent class at fainter
luminosities. Disc-dominated galaxies comprise 10-14\% of the red sequence
population in the low (WINGS) and high (HCS) redshift samples, although their
fraction increases up to 40\% at $0.4 < z < 0.8$ (EDisCS). We find a 20\%
increase in the fraction of S0 galaxies from $z \sim 1.5$ to $z \sim 0.05$ on
the red sequence. These results suggest that elliptical and S0 galaxies follow
different evolutionary histories and, in particular, that S0 galaxies result,
at least at intermediate luminosities ($-22.0 < M_V < -20.0$), from the
morphological transformation of quiescent spiral galaxies.
