  Generalizing the decomposition of a connected planar graph into a tree and a
dual tree, we prove a combinatorial analog of the classic Helmholz-Hodge
decomposition of a smooth vector field. Specifically, we show that for every
polyhedral complex, $K$, and every dimension, $p$, there is a partition of the
set of $p$-cells into a maximal $p$-tree, a maximal $p$-cotree, and a
collection of $p$-cells whose cardinality is the $p$-th Betti number of $K$.
Given an ordering of the $p$-cells, this tri-partition is unique, and it can be
computed by a matrix reduction algorithm that also constructs canonical bases
of cycle and boundary groups.
