On BC-trees and BC-subtrees

TL;DR

This paper explores BC-trees, a special class of trees where the distance between leaves is even, providing algorithms for their construction, introducing BC-subtrees, and analyzing extremal properties and the structure's middle part.

Contribution

It introduces BC-trees and BC-subtrees, offers constructive algorithms for BC-trees with specified parameters, and investigates extremal and structural properties of these trees.

Findings

Algorithms for constructing BC-trees with given order and leaves

Introduction of BC-subtrees and their properties

Analysis of extremal counts and the 'middle part' of BC-trees

Abstract

A BC-tree (block-cutpoint-tree) is a tree (with at least two vertices) where the distance between any two leaves is even. Motivated from the study of the "core" of a graph, BC-trees provide an interesting class of trees. We consider questions related to BC-trees as an effort to make modest progress towards the understanding of this concept. Constructive algorithms are provided for BC-trees with given order and number of leaves whenever possible. The concept of BC-subtrees is naturally introduced. Inspired by analogous work on trees and subtrees, we also present some extremal results and briefly discuss the "middle part" of a tree with respect to the number of BC-subtrees.