  We give a geometric realization of the polyhedra governed by the structure of
associative algebras with co-inner products, or more precisely, governed by
directed planar trees. Our explicit realization of these polyhedra, which
include the associahedra in a special case, shows in particular that these
polyhedra are homeomorphic to balls. We also calculate the number of vertices
of the lowest generalized associahedra, giving appropriate generalizations of
the Catalan numbers.
