We examined the morphology-density relations for galaxy samples selected by luminosity and by mass in each of five massive X-ray clusters from z=0.023 to 0.83 for 674 spectroscopically-confirmed members. Rest-frame optical colors and visual morphologies were obtained primarily from Hubble Space Telescope images. Morphology-density relations (MDR) are derived in each cluster from a complete, luminosity-selected sample of 452 galaxies with a magnitude limit M_V < M^{*}_{V} + 1. The change in the early-type fraction with redshift matches previous work for massive clusters of galaxies. We performed a similar analysis, deriving MDRs for complete, mass-selected samples of 441 galaxies with a mass-limit of 10^{10.6} M_{sun}. Our mass limit includes faint objects, the equivalent of =~1 mag below L^{*} for the red cluster galaxies, and encompasses =~70% of the stellar mass in cluster galaxies. The MDRs in the mass-selected sample at densities of Sigma > 50 galaxies Mpc^{-2} are similar to those in the luminosity-selected sample but show larger early-type fractions. However, the trend with redshift in the fraction of elliptical and S0 galaxies with masses > 10^{10.6} M_{sun} differs significantly between the mass- and luminosity-selected samples. The clear trend seen in the early-type fraction from z=0 to z=~ 0.8 is not found in mass-selected samples. The early-type galaxy fraction changes much less, and is consistent with being constant at 92% +/- 4% at Sigma> 500 galaxies Mpc^{-2} and 83 +/- 3% at 50 < Sigma < 500 galaxies Mpc^{-2}. This suggests that galaxies of mass lower than > 10^{10.6} M_{sun} play a significant role in the evolution of the early-type fraction in luminosity-selected samples. (Abstract abridged)
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