A *binary fractal tree* is defined recursively by symmetric binary
branching. The trunk of length 1 splits into two branches of length
*r*, each making an angle q with the
direction of the trunk. Both of these branches divides into two branches
of length *r*^{2}, each making an angle q with the direction of its parent branch.
Continuing in this way for infinitely many branchings, the tree is the set
of branches, together with their limit points, called *branch tips*.

This study is an elaboration of Chapter 16 of [FGN].

In the obvious way, each branch is determined by a string of symbols
*L* and *R* specifying the choice of direction taken along the
tree to reach the branch. A branch determined by a string of *n*
symbols has length *r*^{n}; a branch tip is determined by an
infinite string of symbols. Most of the analysis in
[FT] results from converting eventually
periodic symbol strings of branch tips into geometric series for the
*x* and *y* coordinates of the branch tips, and making
appropriate interpretations.

For example, in the tree on this page the branch tip marked * can be reached in two ways

andLRRRRLRLRLRLRLR...

RLLLLRLRLRLRLRL...

Consequently, for both sequences, the corresponding branch tips have
*x*-coordinate 0. With simple trigonometry these sequences are
converted into geometric series, giving *r* as a function of q. Details can be found in
[FT].

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