A bijective proof of the Shor recurrence

In an approach to the Cayley formula for counting trees, Shor discovered a refined recurrence relation concerning the number of improper edges. Chen and the author gave a bijection for the Shor recurrence based on the combinatorial interpretations of Zeng, answering a question of Shor. In this paper, we present a new bijective proof of the Shor recurrence by applying Shor's formula for counting forests of rooted trees with roots 1,…,r and with a given number of improper edges.