On the classification of fractal square dendrites

TL;DR

This paper classifies fractal square dendrites based on their self-similar boundary types and main tree structures, identifying 5 boundary types and 16 main tree configurations.

Contribution

It provides a comprehensive classification of fractal square dendrites by analyzing boundary points and main tree structures, revealing 5 boundary types and 16 main tree classes.

Findings

Self-similar boundary of fractal square dendrites can be of 5 types.

Main trees of these dendrites belong to 7 classes.

There are 16 possible main tree types based on boundary placement and order.

Abstract

We consider the classification of fractal square dendrites $K$ based on the types of the self-similar boundary $\partial K$ and the main tree $\gamma$ of such dendrites. We show that the self-similar boundary of a fractal square dendrite $K$ may be of 5 possible types and may consist of 3,4 or 6 points. We prove that the main trees of fractal square dendrites belong to 7 possible classes. Bearing in mind the placement and orders of the points of $\partial K$ with respect to the main tree $\gamma$, this results in 16 possible types of main trees for non-degenerate fractal square dendrites.