The labels in this tree are 2×2 matrices and as such are representations of a geometric structure as explained in “Matrix Gateway to Geometric Algebra, Spacetime, and Spinors” by G. Sobczyk. There are many other sources for the same information, but his book is concise, accessible for an undergraduate in math or physics, and inexpensive.
Clearly, L and R correspond to left and right moves along paths in the binary tree, or a sequence of selections of left or right branches in the tree.
The dyadic monoid labels of the infinite binary tree can be viewed as a certain subset of the modular group PSL(2,Z).
21st Century Paradox 