This work investigates the problem of identifying admissible paths in a tree T to provide a complete characterization of a reduced Gröbner basis \(\mathcal {G}\) of the binomial edge ideal of T. To this end, we develop efficient algorithms that compute all admissible paths in G without exhaustively generating and checking every possible path. We explore three labeling strategies based on classical graph traversal methods, Depth-First Search (DFS), Breadth-First Search (BFS), and inorder traversal for rooted binary trees. For each labeling, we find admissible paths, present optimized algorithms to compute them, and analyze their computational complexity. In the DFS context, we demonstrate that any tree T is m-closed for some \(m \le \textrm{rad}(T) + 2\) , where \(\textrm{rad}(T)\) is the radius of the tree. Via inorder traversal, we show that every rooted binary tree of height h is m-closed for some \(m \le h + 1\) . These results contribute to the structural understanding of binomial edge ideals and their Gröbner bases under various graph labelings.
