- Select
**Design\Examples\Symmetric Binary Trees** - Select a branching angle θ between 0 and 180°.
- Select a scaling ratio between 0 and 1. The button labeled "R" will compute the unique scaling ratio for which the corresponding tree is self-contacting.
- 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 if you just want the final result. The two methods will use different IFS systems.
- Click on "Create IFS" when done.
- For best results with the deterministic method, the program will automatically select a vertical line of length 1 along the y-axis for the design polygon and for the initial set in the fractal window. However, you can choose any line segment as the initial set from the Draw/Add Picture menu These choices will allow the program to quickly draw each iteration in the deterministic algorithm using recursion if that choice is checked. There are two coloring schemes possible in this case. "Use IFS Color Scheme" will use the colors of the first two functions in the IFS window for the left and right branches, respectively. Select "Overlay Images" to color the new branches for each iteration in their own color based on the order of the colors in the IFS Palette (see Code\IFS Color Scheme\Edit IFS Palette). Note that gradient coloring and color stealing will not be used here even if checked in the Code\IFS Color Scheme menu. See examples below. Also note that the window scale is chosen to work best with the choice of a vertical line segment for the initial set. If you use other initial line segments, you may need to adjust the window manually to find the best fit.

**Warning**: Attempting to draw too many iterations using recursion may crash the program and/or freeze your computer. There is no way to stop the drawing in the middle of a recursive drawing except by using ctrl-alt-del to kill the application, so be careful using recursion! The program will attempt to calculate the largest number of iterations before new line segments are less than the size of a pixel. Any iterations after that using recursion will not produce any change in the image. - With the random algorithm you have the option of choosing to create the IFS for just the branch tips.
- If the design polygon or initial set is not a line segment, or if the symmetric binary tree design window is closed, then the iterations for the deterministic algorithm will revert to the standard deterministic drawing method and color schemes.
- If you open an IFS file for a symmetric binary tree and choose the deterministic method with an appropriate design polygon, the program will use recursion if you select "Symmetric Binary Tree" from the Design menu and make sure that "Use recursion" is checked. You must leave this window open while drawing the tree, although you may minimize the window to get it out of the way.

45 degrees, Use IFS Color Scheme | 60 degrees, Overlay Colors |

- Select
**Design\Examples\Symmetric Binary Trees** - Select a branching angle between 0 and 180° for the left branches, and a branching angle between 0 and 180° for the right branches.
- Select a scaling ratio between 0 and 1 for the left branches, and a scaling ratio between 0 and 1 for the right branches.
- Pick an algorithm. The deterministic method works best to view the construction of the tree during the iterations. For a non-symmetric tree, the random method will only draw the branch tips.
- Click on "Create IFS" when done.
- Items (6) to (9) above still apply.

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.