  The Chan-Robbins-Yuen polytope can be thought of as the flow polytope of the
complete graph with netflow vector $(1, 0, \ldots, 0, -1)$. The normalized
volume of the Chan-Robbins-Yuen polytope equals the product of consecutive
Catalan numbers, yet there is no combinatorial proof of this fact. We consider
a natural generalization of this polytope, namely, the flow polytope of the
complete graph with netflow vector $(1,1, 0, \ldots, 0, -2)$. We show that the
volume of this polytope is a certain power of $2$ times the product of
consecutive Catalan numbers. Our proof uses constant term identities and
further deepens the combinatorial mystery of why these numbers appear. In
addition we introduce two more families of flow polytopes whose volumes are
given by product formulas.
