Modeling the Circle Arrangement Problem with Tree Diagrams

Can Buyukugurgor; Emre Yavuz

Modeling the Circle Arrangement Problem with Tree Diagrams

Author(s)	Can Buyukugurgor, Emre Yavuz
Country	Turkey
Abstract	This project investigates how tree diagrams can be used to model and analyze a combinatorial circlearrangement problem. The problem concerns the placement of non-negative integers around a circleunder fixed adjacency conditions, and it asks how many valid configurations can be formed for a givennumber of terms. The main aim of the study is not only to solve a particular counting problem, but alsoto evaluate whether tree diagrams can serve as an effective mathematical modeling tool for complexbranching structures.The study began with a literature review on counting problems, mathematical modeling, treediagrams, and Catalan numbers. After the theoretical background had been established, the circlearrangement problem was examined for small values of n, and all valid arrangements were listedexplicitly. These early cases were then transformed into a tree-diagram model, making the structure ofthe problem more visible and easier to analyze. As the number of terms increased, the tree-diagram method allowed the counting process to remain systematic and organized. One of the most important findings of the project is that the sequence obtained for the main version ofthe problem matches the Catalan sequence for the computed values. This observation gives the projectstronger mathematical significance by connecting the circle arrangement problem to a well-known combinatorial structure. In addition, a Python program was written to compute the number of validarrangements for different values of n, making the method more practical and scalable.The project also considers a generalized version of the problem in which the absolute differencebetween neighboring numbers is at most 2. This extension shows that the same modeling logic can beadapted to richer and more complex constraints. Overall, the study demonstrates that tree diagrams arevaluable not only as visual aids, but also as tools for mathematical reasoning, pattern discovery, andcomputational verification.
Keywords	Tree Diagram, Counting Problem, Mathematical Modeling, Catalan Numbers, Combinatorics
Field	Mathematics
Published In	Volume 8, Issue 5, September-October 2026
Published On	2026-09-16

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Modeling the Circle Arrangement Problem with Tree Diagrams

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