Tree structure of a percolating universe

We present a numerical study of topological descriptors of initially Gaussi an and scale-free density perturbations evolving via gravitational instabil ity in an expanding Universe. The measured Euler number of the excursion se t at the percolation threshold, delta (c), is positive and nearly equal to the number of isolated components, suggesting that these structures are tre es. Our study of critical point counts reconciles the clumpy appearance of the density field at delta (c) with measured filamentary local curvature. I n the Gaussian limit, we measure \delta (c)\ > sigma where sigma (2) is the variance of the density field.