On a variant of $k$-plane trees

Source abstract

In this paper, we introduce a class of plane trees whose vertices receive labels from the set $\{1, 2, \dots, k\}$ such that the sum of labels of adjacent vertices does not exceed $k + 1$ and all vertices of label $1$ are always on the left of all other vertices. Using generating functions, we enumerate these trees by number of vertices and label of the root, root degree, label of the eldest or youngest child of the root and forests. Accepted: 9 February 2025