Inequalities between Gamma-Polynomials of Graph-Associahedra

We prove a conjecture of Postnikov, Reiner and Williams by defining a partial order on the set of tree graphs with n vertices that induces inequalities between the γ-polynomials of their associated graph-associahedra. The partial order is given by relating trees that can be obtained from one another by operations called tree shifts. We also show that tree shifts lower the γ-polynomials of graphs that are not trees, as do the flossing moves of Babson and Reiner.