to determine the completeness of the catalogue as a function of richness. The lines in this figure join the mean redshift value of each bin. At $z\leq0.35$, the galaxy catalogue is volume limited and the survey depth is brighter than the fiducial luminosity cut of $0.2 \, L^{\star}$ (therefore $S(z)=1$). At these redshifts the cluster catalogue is 50 per cent complete down to clusters with $M_{h} \gtrsim 0.2 \times 10^{15} \, \mathrm{M_{\odot}}$. At $z>0.35$, however, the magnitude limit of the survey causes only the most massive clusters to be detected. This causes the richness detection threshold to increase with redshift.

In this work, we restrict our analysis to only consider clusters in the volume limited sample at $z\leq0.35$. We further only consider redMaPPer clusters with halo masses above the 50 per cent $M_{h}$ completeness limit. This is done to maximize the number of clusters in the evolutionary sequence (as discussed in Section \[sec:evol\_seq\_clusters\]).

![image](f_2)

### BCGs - Identification and stellar masses {#sec:BCG_selection}

In our work we select the first CG candidate (the galaxy with the highest $P_{\rm{CEN}}$) in each cluster as the BCG candidate.

The redMaPPer catalogue does not supply any stellar mass $(M_{\star})$ estimates for the galaxies. We determine 