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