But wait! There's more!

At Hat tilings via HTPF equivalence, we've got a lot more code and patterns. For example, each hat tile cuts up a hexagon in the hexagonal grid into 2 or 3 pieces. The division orientation is either Y or upside-down Y. If you color the Y divisions red, you get the pattern below. Code for that and many other items at the link.

Can the single tile be generated by some kind of morphing of Penrose's kite and dart?

You could try mapping the "drafter" Triangle into Penrose triangles and hope that the angles / matching rules work out. But I would guess no because how difficult (impossible?) it would be for patterns fitting on a hexagonal grid to ever allow five-fold centers of symmetry.

As far as structural analysis is concerned, Figure 2.12 could be a good place to start. It looks a bit like this space filling tree dragon we discovered at winter school:

In the future we'll probably get a definition of the tree structure itself, separate from definition in tiling context, but who will be the first to find it?