  This work presents a data-driven reduced-order modeling framework to
accelerate the computations of $N$-body dynamical systems and their pair-wise
interactions. The proposed framework differs from traditional acceleration
methods, like the Barnes-Hut method, which requires online tree building of the
state space, or the fast-multipole method, which requires rigorous $a$ $priori$
analysis of governing kernels and online tree building. Our approach combines
Barnes-Hut hierarchical decomposition, dimensional compression via the
least-squares Petrov-Galerkin (LSPG) projection, and hyper-reduction by way of
the Gauss-Newton with approximated tensor (GNAT) approach. The resulting
$projection-tree$ reduced order model (PTROM) enables a drastic reduction in
operational count complexity by constructing sparse hyper-reduced pairwise
interactions of the $N$-body dynamical system. As a result, the presented
framework is capable of achieving an operational count complexity that is
independent of $N$, the number of bodies in the numerical domain. Capabilities
of the PTROM method are demonstrated on the two-dimensional fluid-dynamic
Biot-Savart kernel within a parametric and reproductive setting. Results show
the PTROM is capable of achieving over 2000$\times$ wall-time speed-up with
respect to the full-order model, where the speed-up increases with $N$. The
resulting solution delivers quantities of interest with errors that are less
than 0.1$\%$ with respect to full-order model.
