  We define an n-dimensional polytope Pi_n(x), depending on parameters x_i>0,
whose combinatorial properties are closely connected with empirical
distributions, plane trees, plane partitions, parking functions, and the
associahedron. In particular, we give explicit formulas for the volume of
Pi_n(x) and, when the x_i's are integers, the number of integer points in
Pi_n(x). We give two polyhedral decompositions of Pi_n(x), one related to order
cones of posets and the other to the associahedron.
