Branch-Weight Unique Trees

Abstract

A branch at a vertex x in a tree is a maximal subtree containing x as an endvertex. The branch-weight of x is the maximum number of edges in any branch at x. The branch-weight sequence of a tree is the multiset consisting of the branch-weights of all vertices arranged in nonincreasing order. Non-isomorphic trees may have the same branch-weight sequence. A tree T is said to be branch-weight unique in a family of trees if T is uniquely determined in the family by its branch-weight sequence. A spider is a tree in which exactly one vertex has degree exceeding two. It is known that spiders are branch-weight unique in the family of spiders but not in the family of all trees. In this study, a necessary and sufficient condition is obtained whereby a spider may be branch-weight unique in the family of all trees. Moreover, two types of trees are proposed to be branch-weight unique in the family of all trees.

Keywords

• branch-weight
• branch-weight sequence
• branch-weight unique
• tree
• spider
• 05c05
• 05c60