Enumeration of plane and d-ary tree-like structures

Abstract

Trees are generalized using various approaches such as considering tree-like structures. Some of the tree-like structures are Husimi graphs, cacti and oriented cacti. These graphs have been enumerated according to number of vertices, blocks, block types and degree sequences. Noncrossing and plane counterparts have also been enumerated by number of vertices, blocks and block types. In this paper, we enumerate plane Husimi graphs, cacti and oriented cacti according to root degree, outdegree of a given vertex and outdegree sequence. The d-ary tree like structures are also introduced in this paper and enumerated according to number of vertices, blocks, block types, outdegree sequence and number of leaves.