On this Math Overflow thread, John Mangual describes a construction of a tree based on the Fibonacci word, and asks, "Is there any regularity to the location of the branches"?
The answer appears to be yes. In the following image of the first 500 levels of the tree, the branches are represented by black pixels. The root of the tree is in the upper left. If you cannot see the pixels, you can download the image and enlarge it in an image editor, or use your browser's "zoom" feature.
A nice spiral pattern is evident. I haven't yet attempted to describe the curves seen in the pattern, or even to prove its existence.
Here is a similar plot. This time black pixels denote "1"s. The "jogs" that are visible are located near the branches of the tree.
Here is the code (in perl) I used to produce
the second image (it can be easily modified to produce the first by
modifying the translation from symbols to colors in the subroutine
maketree). The output should be saved as a pbm file, from which it can
be converted to your favorite graphics format using the netpbm tools
that seem to come with most linuxes.
Updated February 19, 2012