  We introduce a "system-wide safety staffing" (SWSS) parameter for multiclass
multi-pool networks of any tree topology, Markovian or non-Markovian, in the
Halfin-Whitt regime. This parameter can be regarded as the optimal reallocation
of the capacity fluctuations (positive or negative) of order $\sqrt{n}$ when
each server pool employs a square-root staffing rule. We provide an explicit
form of the SWSS as a function of the system parameters, which is derived using
a graph theoretic approach based on Gaussian elimination.
  For Markovian networks, we give an equivalent characterization of the SWSS
parameter via the drift parameters of the limiting diffusion. We show that if
the SWSS parameter is negative, the limiting diffusion and the diffusion-scaled
queueing processes are transient under any Markov control, and cannot have a
stationary distribution when this parameter is zero. If it is positive, we show
that the diffusion-scaled queueing processes are uniformly stabilizable, that
is, there exists a scheduling policy under which the stationary distributions
of the controlled processes are tight over the size of the network. In
addition, there exists a control under which the limiting controlled diffusion
is exponentially ergodic. Thus we have identified a necessary and sufficient
condition for the uniform stabilizability of such networks in the Halfin-Whitt
regime.
  We use a constant control resulting from the leaf elimination algorithm to
stabilize the limiting controlled diffusion, while a family of Markov
scheduling policies which are easy to compute are used to stabilize the
diffusion-scaled processes. Finally, we show that under these controls the
processes are exponentially ergodic and the stationary distributions have
exponential tails.
