MATHEMATICAL AND GRAPH-THEORETICAL ANALYSIS OF BEHAVIOR TREES FOR NPC MODELING

Year 2026, Volume: 27 Issue: 1 , 111 - 124 , 27.03.2026

Seda Göktepe Körpeoğlu , Melike Gören İrem Arslantürk

https://doi.org/10.18038/estubtda.1778255

https://izlik.org/JA77KH57TM

Abstract

Behavior Trees (BTs) have emerged as a widely adopted method for modeling Non-Player Character (NPC) behavior in digital games, offering modularity, scalability, and flexibility compared to traditional approaches such as Finite State Machines. This paper explores the theoretical underpinnings of BTs and their application in NPC modeling. Behavior trees are formalized using graph theory, propositional logic, and probabilistic models, and their computational complexity is analyzed to position them within the broader mathematical framework of decision-making systems. Furthermore, practical examples are presented to demonstrate the advantages of BTs in creating adaptive and realistic NPC behaviors. The results highlight that BTs not only provide practical benefits for game developers but also offer a rigorous mathematical structure for analyzing decision-making models, enabling comparison with alternative approaches such as Markov Decision Processes and reinforcement learning.

Keywords

Artificial intelligence , Digital games , Behavior trees , Mathematical modeling , Game NPCs

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