The answer to the question "Is there any regularity to the location of the branches?" appears to be yes. Here is a diagram of the first 500 branches (a black pixel denotes a branch, and a white pixel denotes no branch, with the root in the upper-left-hand corner just like in your diagram). The regularity is apparent. You might need to zoom in if the pixels are too small.
Of course, I say "appears to be" because I haven't yet proved that the pattern continues, or even described precisely what the pattern is.
P.S.: I would have left this as a comment, since I doubt it qualifies as a full "answer", but apparently I need more reputation to do that. My apologies.
