Factoring, into edge transposition of a tree, permutations fixing a terminal vertex (Q1279659)

For each edge of a tree \(T\) with vertex set \(X\), we associate an edge-transposition of \(X\) which interchanges the ends of the edge. The set of these transpositions generates the symmetric group on \(X\). Identities for these transpositions, including a set of defining relations, are given. The author shows that any minimum length factorization into edge-transpositions, of a permutation of \(X\) fixing a leaf of \(T\), does not involve the unique edge of \(T\) incident with the leaf. This answers a question of \textit{T. P. Vaughan} [J. Comb. Math. Comb. Comput. 10, 65-81 (1991; Zbl 0763.05005)].