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.