How to Create a Symmetric Binary Tree

1. Select File\New\IFS (F5).
2. Select Design\Examples\Symmetric Binary Tree
3. Select a branching angle θ between 0 and 180 degrees.
4. Select a scaling ratio between 0 and 1. The button labeled "R" will compute the unique scaling ratio for which the correspoinding tree is self-contacting.
5. Pick an algorithm. The deterministic method works best to view the construction of the tree during the iterations. The random method draws the tree more quickly. The two methods will use different IFS systems.
6. Click on "Create IFS" when done. If the preview window is open you can see a rough approximation of the final image (approximately 2000 points plotted).

The deterministic method will select an initial polygon consisting of a vertical line of length 1 along the y-axis. Each time you select draw, the program will draw the next set of branches for the tree. The following image shows the self-contacting binary tree for θ = 60 degrees after 12 iterations.

If you select the deterministic method in the Symmetric Binary Tree dialog box, and then choose Random from the Draw menu, you can draw just the set of branch points for the tree.

For more information, see the website Geometry of Self-Contacting Binary Fractal Trees by Benoit Mandelbrot and Michael Frame or the article "Finding Gold in the Forest: Self-contacting Symmetric Binary Fractal Trees and the Golden Ratio", by Tara Taylor, St. Francis Xavier University. Further details are provided in Self-Contacting Binary Trees by Don West, Department of Mathematics, SUNY Plattsburgh.