Counting vertices among all higher-dimensional plane trees

In this paper, we study the enumeration of vertices in d-dimensional plane trees with respect to their levels and degrees. This class of trees generalizes both ordinary plane trees and noncrossing trees. Our approach builds upon a decomposition framework that extends the butterfly decomposition of plane trees introduced by Chen, Li and Shapiro, as well as that of noncrossing trees studied by Oduol and Okoth. We derive both explicit and asymptotic formulas for the enumeration of vertices, eldest children, first children, non-first children and non-leaves at specified levels and degrees. The results are obtained through a combination of generating function techniques, refined butterfly decompositions and bijective methods. This work extends previous enumeration results on ordinary plane trees and noncrossing trees and provides new insights into the combinatorial structure of their higher-dimensional analogues.

کلیدواژه ها:

level ، degree ، eldest child ، first child ، leaf