[205guy]

I believe that the "direct" path of one color and "alternating" path are just a choice of the graphing and not dictated by the data. If you make different node selections, you could make the reverse. Same for the layout of the nicely ordered trees and those that seem more chaotic--it's just a matter of adjusting the layout of the nodes to determine the angles of the edges.

[lmilcin]

I don't mean the order, but the number of paths with given number of nodes of given color. This is independent on the chosen order or layout of the graph.