Tree-like Properties of Cycle Factorizations

We provide a bijection between the set of factorizations, that is, ordered (n− 1)-tuples of transpositions in Sn whose product is (12...n), and labelled trees on n vertices. We prove a refinement of a theorem of Dénes [3] that establishes new tree-like properties of factorizations. In particular, we show that a certain class of transpositions of a factorization correspond naturally under our bijection to leaf edges of a tree. Moreover, we give a generalization of this fact.