  Symmetric edge polytopes of graphs and root polytopes of semi-balanced
digraphs are two classes of lattice polytopes whose $h^*$-polynomials have
interesting properties and generalize important graph polynomials. For both
classes of polytopes there are large, natural classes of dissections into
unimodular simplices. These are such that the simplices correspond to certain
spanning trees.
  We show that for any ``spanning tree dissection'' of the symmetric edge
polytope of a graph, or the root polytope of a semi-balanced digraph, the
$h^*$-polynomial of the polytope can be computed as a generating function of
certain activities of the corresponding spanning trees. Apart from giving
simple and flexible algorithms for computing these polynomials, our results
also reveal that all dissections in question are surprisingly similar to each
other: It turns out that the distributions of many statistics of spanning tree
dissections are in fact independent of the actual dissection.
