  We study random typical minimal factorizations of the $n$-cycle, which are
factorizations of $(1, \ldots,n)$ as a product of $n-1$ transpositions, chosen
uniformly at random. Our main result is, roughly speaking, a local convergence
theorem for the trajectories of finitely many points in the factorization. The
main tool is an encoding of the factorization by an edge and vertex-labelled
tree, which is shown to converge to Kesten's infinite Bienaym\'e-Galton-Watson
tree with Poisson offspring distribution, uniform i.i.d. edge labels and vertex
labels obtained by a local exploration algorithm.
