  We study the geometry of varieties parametrizing degree d rational and
elliptic curves in P^n intersecting fixed general linear spaces and tangent to
a fixed hyperplane H with fixed multiplicities along fixed general linear
subspaces of H. As an application, we derive recursive formulas for the number
of such curves when the number is finite. These recursive formulas require as
``seed data'' only one input: there is one line in P^1 through two points.
These numbers can be seen as top intersection products of various cycles on the
Hilbert scheme of degree d rational or elliptic curves in P^n, or on certain
components of $\mbar_0(P^n,d)$ or $\mbar_1(P^n,d)$, and as such give
information about the Chow ring (and hence the topology) of these objects. The
formula can also be interpreted as an equality in the Chow ring (not
necessarily at the top level) of the appropriate Hilbert scheme or space of
stable maps. In particular, this gives an algorithm for counting rational and
elliptic curves in P^n intersecting various fixed general linear spaces. (The
genus 0 numbers were found earlier by Kontsevich-Manin, and the genus 1 numbers
were found for n=2 by Ran and Caporaso-Harris, and independently by Getzler for
n=3.)
