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Path length and height in asymmetric binary branching trees. (English) Zbl 1154.28301

This paper explores geometric properties of asymmetric binary branching trees, which are self-similar fractal sets as well. Its main achievements are: 1. two equations describing the lengths of some infinite periodic paths, and a significant prerequisite for a pair of periodic opposite paths to be equal. 2. an inequality determining if the \(n\)th branch of greater height is a left or a right one, and a formula for calculating the height of the tree. Relevant proofs are presented by using geometric series, trigonometric identities and the self-similarity in fractal trees. The authors also investigate in this paper methods of drawing and analyzing asymmetric trees in Mathematica, and how to use the package created for this study. Two affine transformations are provided for generating pictures of fractal trees. The proposed open questions in relation to space-filling curves are also worth for a further study.

MSC:

28A80 Fractals
28A75 Length, area, volume, other geometric measure theory
68N30 Mathematical aspects of software engineering (specification, verification, metrics, requirements, etc.)
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