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A fractal tree is “grown” by repeatedly replacing each branch with smaller branches:

The total number of branches and the total length of all of the branches follow geometric series.

We can use Google Sketchup, a free 3D modelling tool, to create fractal trees. We start with the basic design of a branch on the left. We make two copies and resize to 70% (for example) of the original size. We then rotate them both and position them, as in the diagram on the right. We then group these three pieces together. This will be our building block (the diagram on the right that is):

Note that the circle and the line make it easier to position the branches in Google Sketchup. Just drag the mid-point of the base of each of the small branches to the end of the line inside the circle on the large branch. Also, when rotating the branches you can use the circle and line as reference points.

Make two copies of the design on the right which are both 70% smaller. Replace the two branches of the large design with the two smaller copies. Your tree should now have grown by four extra branches.

Keep repeating the previous step again and again, each time making only two copies and replacing the branches on the larger design with the two copies. This is the easiest way to grow the tree. See the attached Google Sketchup file for an example.

Experiment creating different trees by changing the scale of the new copies, the angle of each branch and the initial number of branches (this example has two branches initially).

Calculate the total number of branches and their total length after n iterations…