Trees associated with the Motzkin numbers
\({\mathcal P}_n\) is the set of all plane rooted trees on \(n+1\) unlabeled vertices with edges oriented from the root. \({\mathcal E}_n\) is the set of trees in \({\mathcal P}_n\) with all vertices with an odd distance to the root having at least two outgoing edges. \({\mathcal M}_n\) is the set of trees in \({\mathcal P}_n\) with no vertex having more than two outgoing edges. The Motzkin number \(M_n\) is the size of set \({\mathcal M}_n\). It is shown that \({\mathcal E}_n\) has also size \(M_n\). The number of trees in \({\mathcal E}_n\) with exactly \(k+1\) vertices that have even distance to the root is shown to be \(C_k\).
scientific article; zbMATH DE number 934997