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.
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