Factoring a Permutation on a Broom

A tree \(T\) consisting of a line with edges \(\{(1, 2), (2, 3), \ldots, (n-1, n)\}\) and with edges \(\{(1, a_1), (1, a_2), \ldots, (1, a_k)\}\) (a star) attached on the left, is called a broom.
The edges of the tree \(T\) are called \(T\)-transpositions. We give an algorithm to factor any permutation \(\sigma\) of \(\{a_1, a_2, \ldots, a_k, 1, 2, \ldots ,n\}\) as a product of \(T\)-transpositions, and prove that the factorization produced by the algorithm has minimal length.