Combinatorial and Probabilistic Formulae for Divided Symmetrization

Divided symmetrization of a function $f(x_1,\dots ,x_n)$ is symmetrization of the ratio $$D{S}_G(f)=\frac{f(x_1,\dots ,x_n)}{\prod (x_i-x_j)},$$ 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 $D{S}_G(f)$ 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.