  We present a numerical study of topological descriptors of initially Gaussian
and scale-free density perturbations evolving via gravitational instability in
an expanding universe. We carefully evaluate and avoid numerical contamination
in making accurate measurements on simulated fields on a grid in a finite box.
Independent of extent of non linearity, the measured Euler number of the
excursion set at the percolation threshold, $\delta_c$, is positive and nearly
equal to the number of isolated components, suggesting that these structures
are trees. Our study of critical point counts reconciles the clumpy appearance
of the density field at $\delta_c$ with measured filamentary local curvature.
In the Gaussian limit, we measure $|\delta_c|> \sigma$ in contrast to widely
held belief that $|\delta_c| \sim \sigma$, where $\sigma^2$ is the variance of
the density field.
