  This paper, and its companion, investigate the evolution of dense stellar
systems due to the influence of two-body gravitational encounters, physical
collisions and stellar evolution. Our goal is the simulation of the densest
centers of galaxies, like M32, which reach stellar densities near a million
solar masses per cubic parsec and which may harbor black holes. These systems
have a different Safronov number (the dimensionless ratio of stellar binding
energy to mean stellar kinetic energy) than globular clusters, substantially
increasing the importance of physical collisions relative to gravitational
encounters. In this paper, we focus only on the gravitational encounters. We
demonstrate, first, that our simulations with small N with a Hernquist tree
code yield results basically in accord with years of effort studying globular
clusters. Second, we investigate (crudely) core collapse in rotating systems
with mass segregation, to separate out the effects purely due to two-body
encounters from those seen in the more complex second paper.
  Consistent with previous studies, we find that systems whose constituent
particles follow the Salpeter initial mass function rapidly undergo core
collapse through Spitzer's mass segregation instability. All rotationally
flattened systems show a decrease in flattening with time, consistent with
Fokker-Planck calculations. An interesting new result concerns the
well-established inability of simulators to identify a static center in their
simulations of collisional systems. We find that the lump of high mass stars
which condenses at the center wanders about the core in Brownian motion.
