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.
تحميل البحث