  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 $\ell$, (ii) non-first-children of degree $k$ at level
$\ell-1$, (iii) leaves having $k-1$ elder siblings at level $\ell$, and (iv)
non-leaves of outdegree $k$ at level $\ell-1$. Our results unite and generalize
several previous works in the literature.
