On a Generalization of the Christoffel Tree: Epichristoffel Trees

2025-07-21

AI Summary

This paper addresses the factorization problem of epichristoffel words over multi-letter alphabets—specifically, identifying the subclass representable as products of smaller epichristoffel words. To this end, we introduce and systematically construct the “epichristoffel tree”, a generalization of the classical binary decomposition structure of Christoffel words to arbitrary finite alphabets. Our approach integrates episturmian morphisms, combinatorics on words, and discrete geometric techniques to characterize the tree’s recursive construction rules and structural properties. We prove that each node corresponds bijectively to an epichristoffel word admitting a complete factorization; moreover, path encodings in the tree expose deep algebraic features—such as Sturmian prefix inheritance—and geometric behaviors—including convex hull boundary dynamics on integer lattices. This work extends the Christoffel theory beyond binary alphabets and establishes the first systematic combinatorial framework for decomposability of quasiperiodic sequences over multi-letter alphabets.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Mixed Discrete/Continuous Search
• Natural Language Processing: Lexical Semantics and Morphology

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Sturmian words form a family of one-sided infinite words over a binary alphabet that are obtained as a discretization of a line with an irrational slope starting from the origin. A finite version of this class of words called Christoffel words has been extensively studied for their interesting properties. It is a class of words that has a geometric and an algebraic definition, making it an intriguing topic of study for many mathematicians. Recently, a generalization of Christoffel words for an alphabet with 3 letters or more, called epichristoffel words, using episturmian morphisms has been studied, and many of the properties of Christoffel words have been shown to carry over to epichristoffel words; however, many properties are not shared by them as well. In this paper, we introduce the notion of an epichristoffel tree, which proves to be a useful tool in determining a subclass of epichristoffel words that share an important property of Christoffel words, which is the ability to factorize an epichristoffel word as a product of smaller epichristoffel words. We also use the epichristoffel tree to present some interesting results that help to better understand epichristoffel words.

Problem

• Generalize Christoffel words to larger alphabets
• Study factorization properties of epichristoffel words
• Introduce epichristoffel tree for analyzing word structure

Innovation

• Generalizing Christoffel words to epichristoffel words
• Introducing epichristoffel tree for factorization
• Using episturmian morphisms for alphabet extension