  We investigate the properties of hard core Bosons in harmonic traps over a
wide range of densities. Bose-Einstein condensation is formulated using the
one-body Density Matrix (OBDM) which is equally valid at low and high
densities. The OBDM is calculated using diffusion Monte Carlo methods and it is
diagonalized to obtain the "natural" single particle orbitals and their
occupation, including the condensate fraction. At low Boson density, $na^3 <
10^{-5}$, where $n = N/V$ and $a$ is the hard core diameter, the condensate is
localized at the center of the trap. As $na^3$ increases, the condensate moves
to the edges of the trap. At high density it is localized at the edges of the
trap. At $na^3 \leq 10^{-4}$ the Gross-Pitaevskii theory of the condensate
describes the whole system within 1%. At $na^3 \approx 10^{-3}$ corrections are
3% to the GP energy but 30% to the Bogoliubov prediction of the condensate
depletion. At $na^3 \gtrsim 10^{-2}$, mean field theory fails. At $na^3 \gtrsim
0.1$, the Bosons behave more like a liquid $^4$He droplet than a trapped Boson
gas.
