  All gravitationally bound clusters expand, due to both gas loss from their
most massive members and binary heating. All are eventually disrupted tidally,
either by passing molecular clouds or the gravitational potential of their host
galaxies. However, their interior evolution can follow two very different
paths. Only clusters of sufficiently large initial population and size undergo
the combined interior contraction and exterior expansion that leads eventually
to core collapse. In all other systems, core collapse is frustrated by binary
heating. These clusters globally expand for their entire lives, up to the point
of tidal disruption.
  Using a suite of direct N-body calculations, we trace the "collapse line" in
r_v-N space that separates these two paths. Here, r_v and N are the cluster's
initial virial radius and population, respectively. For realistic starting
radii, the dividing N-value is from 10^4 to over 10^5. We also show that there
exists a minimum population, N_min, for core collapse. Clusters with N < N_min
tidally disrupt before core collapse occurs. At the Sun's Galactocentric
radius, R_G = 8.5 kpc, we find N_min >~ 300. The minimum population scales with
Galactocentric radius as R_G^{-9/8}.
  The position of an observed cluster relative to the collapse line can be used
to predict its future evolution. Using a small sample of open clusters, we find
that most lie below the collapse line, and thus will never undergo core
collapse. Most globular clusters, on the other hand, lie well above the line.
In such a case, the cluster may or may not go through core collapse, depending
on its initial size. We show how an accurate age determination can help settle
this issue.
