A tree-solid

A tree-solid is a scale-symmetric shape which is a tree (acyclic, no holes) but fills the full area of space.

It is the set complement of a void-tree which is what we usually call fractal trees. So people usually just make fractal trees like the Vicsek fractal:

But making one primarily as a tree-solid makes you consider the shape of the solid regions it is built off. In the case of this post, it is disks.

To make a tree from disks they must overlap, so the shape is an overlapping disk packing, with the overlaps in a tree topology.

If is possible to make it with any intersection angle between the disk and its parent, but I used 90 degrees, which is half way between the two extremes:

You can probably just make out the disks that it is build from.

Another way to arrange part of this structure looks more like the Vicsek fractal:

To see the circles more easily in the first image we can colour them according to which iteration they were on:

Same for the second variant:

which can be rotated:

The reason I made this structure is that I'm looking into whether there is a 3D equivalent. This would be a tree-solid, or visualised as its complement: a void-shell. This is a lot harder to make, and may not be possible.

FYI, the limit of both of these variants as the intersection angle decreases to zero is the non-overlapping disk packing here: