  We investigate the geography of Hilbert schemes parametrizing closed
subschemes of projective space with specified Hilbert polynomials. We classify
Hilbert schemes with unique Borel-fixed points via combinatorial expressions
for their Hilbert polynomials. These expressions naturally lead to an
arrangement of nonempty Hilbert schemes as the vertices of an infinite full
binary tree. Here we discover regularities in the geometry of Hilbert schemes.
Specifically, under natural probability distributions on the tree, we prove
that Hilbert schemes are irreducible and nonsingular with probability greater
than $0.5$.
