The Shape of Symmetric Binary Trees

TL;DR

The paper investigates the top, bottom, and side extremities of self-overlapping symmetric binary fractal trees generated by an angle $\theta$ and scale $r$, using a complex-plane framework with $\alpha=e^{i\theta}$ to express branch-tip coordinates. It defines and analyzes the top via $f_{\theta}(r)=y_k-y_0$ and derives critical radii $r_T$ that mark when non-$LR$ paths overtake the alternating path $(LR)^{\infty}$, providing explicit formulas for $r_T$ in the ranges $90^{\circ}<\theta<120^{\circ}$ and $120^{\circ}<\theta<180^{\circ}$, as well as the behavior as $\theta$ approaches special angles. A parallel bottom analysis yields a critical $r_B$ (including a closed form for $\theta>144^{\circ}$) and shows how the bottom tip can shift away from the $(LR)^{\infty}$ path depending on $\theta$ and $r$, influenced by the sign of $N(\theta)$. Side-extents are characterized by the rightmost tip along $R^k(LR)^{\infty}$, yielding a closed form $x_{\max}=\dfrac{r\sin\theta}{1-r^2}$ for $\theta\ge90^{\circ}$, with potential larger tips arising from deeper $R^m(LR)^{\infty}$ paths when $m\theta\ge450^{\circ}$. Overall, the work maps how overlapping changes which tip attains the top, bottom, and side extremes across parameter regimes, and provides explicit criteria to predict extreme-tip locations in these fractal trees.

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.

Figures (17)

• Figure 1: Two iterations
• Figure 2: Symmetric binary trees
• Figure 3: Branch tip labeling
• Figure 4: $\theta = 35^\circ, r = 0.6$ (top = 2.6539, side = 1.51964, bottom = 1.15588)
• Figure 5: $R^6(LR)^8$, $\theta = 70^\circ$