  The type A_n full root polytope is the convex hull in R^{n+1} of the origin
and the points e_i-e_j for 1<= i<j <= n+1. Given a tree T on the vertex set
[n+1], the associated root polytope P(T) is the intersection of the full root
polytope with the cone generated by the vectors e_i-e_j, where (i, j) is an
edge of T, i<j. The reduced forms of a certain monomial m[T] in commuting
variables x_{ij} under the reduction x_{ij}x_{jk} -->
x_{ik}x_{ij}+x_{jk}x_{ik}+\beta x_{ik}, can be interpreted as triangulations of
P(T). Using these triangulations, the volume and Ehrhart polynomial of P(T) are
obtained. If we allow variables x_{ij} and x_{kl} to commute only when i, j, k,
l are distinct, then the reduced form of m[T] is unique and yields a canonical
triangulation of P(T) in which each simplex corresponds to a noncrossing
alternating forest. Most generally, the reduced forms of all monomials in the
noncommutative case are unique.
