- Wow, that's great - thanks! So somehow
\graph [layered layout, level distance=20mm]
created the spacings of the entries within each row? I'm wondering why the space between 10000 and 01000 is so much bigger than the other spaces in that row - that's the main thing I'd want to change, along with the symmetrically large space between 01111 and 10111.
I imagine some TikZ expert could figure out the intelligent way to change the labels. I might do it a dumb way myself.