  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 $\gamma$-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
$\gamma$-polynomials of graphs that are not trees, as do the flossing moves of
Babson and Reiner.
