Enumerations of vertices among all rooted ordered trees with levels and degrees

In this paper we enumerate and give bijections for the following four sets of vertices among rooted ordered trees of a fixed size: (i) first-children of degree $k$ at level $\mathrm{\ell }$, (ii) non-first-children of degree $k$ at level $\mathrm{\ell }-1$, (iii) leaves having $k-1$ elder siblings at level $\mathrm{\ell }$, and (iv) non-leaves of outdegree $k$ at level $\mathrm{\ell }-1$. Our results unite and generalize several previous works in the literature.