Counting formulas for weakly labelled plane tree-like structures

Abayo, Sylvester Arthur; Okoth, Isaac Owino; Kasyoki, Donnie Munyao

doi:10.22061/jdma.2025.11860.1123

Counting formulas for weakly labelled plane tree-like structures

Document Type : Full Length Article

Authors

¹ Department of Pure and Applied Mathematics, Maseno University, Maseno, Kenya

² Department of Pure and Applied Mathematics, School of Mathematics, Statistics and Actuarial Science, Maseno University, Maseno, Kenya

³ Department of Pure and Applied Mathematics, Maseno University, Maseno, Kenya.

10.22061/jdma.2025.11860.1123

Abstract

Block graphs have been enumerated by various authors. In this paper, plane tree-like structures in which the blocks are labelled with integers in the set {1,2,...,k} such that the labels of the blocks are non-decreasing from left to right are introduced. These tree-like structures are called weakly labelled k-plane tree-like structures herein. Using symbolic method, generating functions and application of Lagrange-B\"{u}rmann inversion, the structures are counted by number of vertices, blocks, occurrences of labels, root degree and label of the eldest/youngest block child of the root, number of leaves, forests and outdegree sequence.

Graphical Abstract