  We have obtained images of the Trapezium Cluster (140" x 140"; 0.3 pc x 0.3
pc) with the Hubble Space Telescope Near-Infrared Camera and Multi-Object
Spectrometer (NICMOS). Combining these data with new ground-based K-band
spectra (R=800) and existing spectral types and photometry and the models of
D'Antona & Mazzitelli, we find that the distributions of ages of comparable
samples of stars in the Trapezium, rho Oph, and IC 348 indicate median ages of
\~0.4 Myr for the first two regions and ~1-2 Myr for the latter. The low-mass
IMFs in these sites of clustered star formation are similar over a wide range
of stellar densities and other environmental conditions. With current data, we
cannot rule out modest variations in the substellar mass functions among these
clusters. We then make the best estimate of the true form of the IMF in the
Trapezium by using the evolutionary models of Baraffe et al. and an empirically
adjusted temperature scale and compare this mass function to recent results for
the Pleiades and the field. All of these data are consistent with an IMF that
is flat or rises slowly from the substellar regime to about 0.6 Msun, and then
rolls over into a power law that continues from about 1 Msun to higher masses
with a slope similar to or somewhat larger than the Salpeter value of 1.35. For
the Trapezium, this behavior holds from our completeness limit of ~0.02 Msun
and probably, after a modest completeness correction, even from 0.01-0.02 Msun.
These data include ~50 likely brown dwarfs. We test the predictions of theories
of the IMF against various properties of the observed IMF.
