Shahid Rajaee Teacher Training University Journal of Discrete Mathematics and Its Applications 2981-0809 10 3 2025 09 01 Counting formulas for weakly labelled plane tree-like structures 243 261 2370 10.22061/jdma.2025.11860.1123 EN Sylvester Arthur Abayo Department of Pure and Applied Mathematics, Maseno University, Maseno, Kenya Isaac Owino Okoth Department of Pure and Applied Mathematics, Maseno University, Maseno, Kenya Donnie Munyao Kasyoki Department of Pure and Applied Mathematics, Maseno University, Maseno, Kenya Journal Article 2025 03 16 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. https://jdma.sru.ac.ir/article_2370_cf107bb21068c50136a77b7750750c65.pdf