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 Refined enumeration of vertices among all rooted ordered $d$-trees Bull. Korean Math. Soc. 2019 Vol. 56, No. 4, 899-910 https://doi.org/10.4134/BKMS.b180661Published online July 31, 2019 Sangwook Kim, Seunghyun Seo, Heesung Shin Chonnam National University; Kangwon National University; Inha University Abstract : In this paper, we enumerate the cardinalities for the set of all vertices of outdegree $\ge k$ at level $\ge \ell$ among all rooted ordered $d$-trees with $n$ edges. Our results unite and generalize several previous works in the literature. Keywords : rooted ordered $d$-tree, lattice path, bijection, degree, level MSC numbers : 05A15 Full-Text :