Indian Journal of Science and Technology

Abstract

Objective: This study investigated the properties of fuzzy binaries, a type of rooted trees, and explored their implications in different domains. Method: This investigation included analyzing the structural properties of fuzzy binary tree, such as the number of vertices, the height, and the degree to which edges were assigned membership. Theorem: Combinatorial analysis and mathematical induction were used to derive the relation between the number of vertex and height of a fuzzy binary tree based on its internal vertex. The conditions that allow complete graphs to form within fuzzy binary tree were also determined by mathematical analysis and graph theory. Findings: The study found that the internal vertices of a fuzzy binary tree have a strong influence on the number of vertices and height. A theorem was developed that states a fuzzy binary tree full with ”i” terminal vertices will have (i+1), total vertices, and (i+1), terminal vertices. A fuzzy binary tree can form a complete graph under certain conditions. Novelty: This research provides novel insights into fuzzy binary trees’ structural properties. The derived theorem establishes a fundamental relation between the number and height of a fuzzy binary tree. This provides valuable insight into the complexity and organization of these trees. The identification of the conditions that lead to a complete graph in fuzzy binary tree further improves our understanding and appreciation of their comprehensiveness. These findings provide new perspectives on the application of fuzzy binary tree in different fields.

Keywords: Fuzzy Binary Trees; Rooted Tree; Degrees of Membership; Structural Characteristics; Complete Graphs; Image Processing; Full Binary Tree; Terminal Vertices; Internal Vertices; Membership Function; Fuzzy Set Theory; Graph Theory; Data Mining