Combinatorial and probabilistic formulae for divided symmetrization

Divided symmetrization of a function f(x1,…,xn) is symmetrization of the ratio DSG(f)=f(x1,…,xn)∏(xi−xj),where the product is taken over the set of edges of some graph G. We concentrate on the case when G is a tree and f is a polynomial of degree n−1, in this case DSG(f) is a constant function. We give a combinatorial interpretation of the divided symmetrization of monomials for general trees and probabilistic game interpretation for a tree which is a path. In particular, this implies a result by Postnikov originally proved by computing volumes of special polytopes, and suggests its generalization.