  We study a two-parameter generalization of the Catalan numbers: $C_{d,p}(n)$
is the number of ways to subdivide the $d$-dimensional hypercube into $n$
rectangular blocks using orthogonal partitions of fixed arity $p$. Bremner \&
Dotsenko introduced $C_{d,p}(n)$ in their work on Boardman--Vogt tensor
products of operads; they used homological algebra to prove a recursive formula
and a functional equation. We express $C_{d,p}(n)$ as simple finite sums, and
determine their growth rate and asymptotic behaviour. We give an elementary
proof of the functional equation, using a bijection between hypercube
decompositions and a family of full $p$-ary trees. Our results generalize the
well-known correspondence between Catalan numbers and full binary trees.
