Tripods I

With all the emerging wildflowers, I have been using my tripod a lot lately, and this has led to today’s puzzle. We are going to color perfectly height balanced trivalent trees, like so:

This is of course too easy as it stands, so we have to impose restrictions. Today, I will insist that all tripods in your coloring look like this:

So a tripod is a perfectly height balanced trivalent tree of height 2, technically speaking. A quick inspection shows that the example above is of this sort. There still are many many such colorings, given the symmetries of the tree, and we’ll need further constraints for today’s puzzle. Before we get there, I have a few questions:

• The coloring above uses one color 21 times and the other color 25 times. Is this always the case?
• Is there always a single colored path from leaf to leaf through the center of the tree?
• How many different colorings exist if you disregard symmetries?

Above is a simple version of today’s puzzle. On the left, you see a partially colored tree. On the right there is a completed coloring, following the rules that all tripods in the tree must be colored as the two tripods above. In this case the solution is unique, as also in the puzzle below:

Enjoy. More tripods puzzles next week…