| dc.description.abstract | Trees are connected graphs which do not have loops, multiple edges and cycles. A variety of trees such as binary trees, ordered trees, d-ary trees, Cayley trees and noncrossing trees have been studied at length. Tree-like structures such as cacti and Husimi graphs have the properties of trees where we consider blocks of the structures instead of vertices. Plane Husimi graphs, plane cacti and plane oriented cacti have been enumerated with regards to leaves, number of blocks and block types. However, there is no literature on the study of plane tree-like structures according to root degree and degree sequence. Moreover, d-ary tree-like structures have not been enumerated at all . In this work, we have enumerated plane Husimi graphs, plane cacti and plane oriented cacti according to the degree of the root and outdegree sequences. We have also enumerated bicoloured plane tree-like structures with regards to number of vertices, blocks and block types. Finally, we have introduced and enumerated d-ary Husimi graphs, cacti and oriented cacti with given indegree sequence, number of leaves, blocks and block types. To obtain our results we have used symbolic method to obtain generating functions for tree-like structures, used Lagrange Inversion formula and Lagrange Burmann to extract the coefficients of the variables in the generating¨ functions and in some instance, we constructed a bijection. The results of this study will add to literature in this area of study. | en_US |
