  Rigid tree honeycombs were introduced by Knutson, Tao, and Woodward and they
were shown by Dykema, Collins, Timotin, and the authors to be sums of extreme
rigid honeycombs, with uniquely determined summands up to permutations. Two
extreme rigid honeycombs are essentially the same if they have proportional
exit multiplicities and, up to this identification, there are countably many
equivalence classes of such honeycombs. We describe two ways to approach the
enumeration of these equivalence classes. The first method produces a (finite)
list of all rigid tree honeycombs of fixed weight by looking at the locking
patterns that can be obtained from a certain quadratic Diophantine equation.
The second method constructs arbitrary rigid tree honeycombs from rigid
overlays of two rigid tree honeycombs with strictly smaller weights. This
allows, in principle, for an inductive construction of all rigid tree
honeycombs starting with those of unit weight. We also show that some rigid
overlays of two rigid tree honeycombs give rise to an infinite sequence of
rigid tree honeycombs of increasing complexity but with a fixed number of
nonzero exit multiplicities. This last result involves a new
inflation/deflation construction that also produces other infinite sequences of
rigid tree honeycombs.
