A generating tree with a single label for permutations avoiding the vincular pattern 1-32-4

In this paper we continue the study of permutations avoiding the vincular pattern $1-32-4$ by constructing a generating tree with a single label for these permutations. This construction finally provides a clearer explanation of why a certain recursive formula found by Callan actually counts these permutations, insofar as this formula was originally obtained only as a consequence of a very intricated bijection with a certain class of ordered rooted trees. This responds to a theoretical issue already raised by Duchi, Guerrini and Rinaldi. As a byproduct, we also obtain an algorithm to generate all these permutations and we refine their enumeration according to a simple statistic, which is the number of right-to-left maxima to the right of 1.