The Shape of Symmetric Binary Trees

TL;DR

This paper investigates the geometric properties of symmetric binary trees, especially focusing on the height of branch tips and conditions under which self-overlapping trees differ from non-overlapping ones, extending previous theoretical results.

Contribution

It constructs examples of self-overlapping symmetric binary trees for almost all angles and analyzes conditions affecting their geometric structure.

Findings

Most angles produce self-overlapping trees with different height properties.

Constructed examples show deviations from previous theorems for certain configurations.

Conditions identified for when sides and bottom of trees differ from non-overlapping cases.

Abstract

Mandelbrot and Frame studied the geometry of self-contracting symmetric binary trees in which they stated that the height of such trees occurred at the branch tip of the path consisting of branches that alternate left and right. Taylor proved that this happens for both self-avoiding as well as self-contacting symmetric binary trees (if we ignore the height of the trunk and just consider the branch tips). In his commentary on the work by Mandelbrot and Frame, West gave an example of a self-overlapping tree in which this alternating left-right path does not give the highest point of the tree, and said that more analysis was needed. In this paper we show how such examples can be constructed for all but a countable number of angles. We also investigate the conditions for when the sides and bottom of a self-overlapping symmetric binary tree differ from what happens with self-avoiding and self-contacting trees.