We present a detailed morphological analysis of the galaxy populations in the first two clusters to be completed in an extensive observational study of nine high-redshift clusters of galaxies. These two clusters, Cl 0023+0423 and Cl 1604+4304, are at redshifts of z = 0.84 and z = 0.90, respectively. The morphological studies are based on high angular resolution imagery taken with Wide Field Planetary Camera 2 aboard the Hubble Space Telescope. These data are combined with deep, ground-based BVRI photometry and spectra taken with the Keck 10 m telescopes. The morphological classifications presented in this paper consist of two parts. First, we provide a quantitative description of the structural properties of ~600 galaxies per cluster field using the Medium Deep Survey automated data reduction and object classification software. This analysis includes the galaxy position, photometry, and best-fit bulge+disk model. Second, for the brightest subsample of ~200 galaxies per cluster field, we provide a more detailed morphological description through a visual classification based on the revised Hubble classification scheme. Based on these classifications, we have examined the general relation between galaxy morphology and other photometric and spectral properties. We find that, as expected, the elliptical and S0 galaxies are redder, on average, than are the spiral and irregular galaxies. In addition, there is a strong correlation between morphology and spectral type. Of the galaxies that are visually classified as ellipticals, the majority show K star absorption spectra that are typical of nearby, red early-type galaxies; however, a few are actually blue compact galaxies with spectra characterized by fairly strong, narrow emission lines. Normal late-type galaxies typically have spectra with blue colors and [O II] emission, while the presence of strong star formation features, such as extremely high equivalent width [O II], Hβ, and/or [O III] emission, is always accompanied by peculiar morphologies that suggest recent mergers or interactions. We have used the statistical distributions of cluster galaxy morphologies to probe the overall morphological composition of these two systems. This analysis reveals that the two clusters contain very different galaxy populations. Cl 0023+0423 has a galaxy population that is more similar to groups of galaxies and the field. This system is almost completely dominated by spiral galaxies. Cl 1604+4304, however, has a morphological composition that is more typical of a normal, present-day cluster; early-type galaxies make up ~76% of all galaxies brighter than MV = -19.0 + 5 log h in the central ~0.5 h-1 Mpc. The ratio of S0 galaxies to elliptical galaxies in this cluster is 1.7 ± 0.9, consistent with local cluster populations. The morphological results support the conclusions of the dynamical analysis presented in the second paper of this series. Cl 0023+0423 consists of two galaxy groups that are separated by ~2900 km s-1 in radial velocity. Cl 1604+4304, on the other hand, has a velocity distribution indicating that it is already well formed and relaxed. The morphological composition, velocity dispersion, and implied mass of the Cl 1604+4304 system are consistent with an Abell richness class 2 or 3 cluster.
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