A remark on the enumeration of rooted labeled trees

Two decades ago, Chauve, Dulucq and Guibert showed that the number of rooted trees on the vertex set $[n+1]$ in which exactly $k$ children of the root are lower-numbered than the root is $\binom{n}{k} \, n^{n-k}$. Here I give a simpler proof of this result.
Comment: LaTex2e, 9 pages. Version 2 contains a Note Added with a quick and elegant proof due to Jiang Zeng. To be published in Discrete Mathematics