  We introduce the notion of "binary" positive and complex geometries, giving a
completely rigid geometric realization of the combinatorics of generalized
associahedra attached to any Dynkin diagram. We also define open and closed
"cluster string integrals" associated with these "cluster configuration
spaces". The binary geometry of type ${\cal A}$ gives a gauge-invariant
description of the usual open and closed string moduli spaces for tree
scattering, making no explicit reference to a worldsheet. The binary geometries
and cluster string integrals for other Dynkin types provide a generalization of
particle and string scattering amplitudes. Both the binary geometries and
cluster string integrals enjoy remarkable factorization properties at finite
$\alpha'$, obtained simply by removing nodes of the Dynkin diagram. As
$\alpha'\to 0$ these cluster string integrals reduce to the canonical forms of
the ABHY generalized associahedron polytopes. For classical Dynkin types these
are associated with $n$-particle scattering in the bi-adjoint $\phi^3$ theory
through one-loop order.
