  Given a finite directed acyclic graph, the space of non-negative unit flows
is a lattice polytope called the flow polytope of the graph. We consider the
volumes of flow polytopes for directed acyclic graphs on $n+1$ vertices with a
fixed degree sequence, with a focus on graphs having in- and out-degree two on
every internal vertex. When the out-degree of the source is three and the
number of vertices is fixed, we prove that there is an interchange operation on
the edge set of these graphs that induces a partial order on the graphs
isomorphic to a Boolean algebra. Further, we prove that as we move up through
this partial order, the volumes of the corresponding flow polytopes weakly
decrease. Finally, we show that each such graph is strongly planar and we
provide an alternative interpretation of our results in the context of linear
extensions for posets that are bipartite non-crossing trees.
