Geodesic
Branch-Weight Unique Trees
Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 2, pp. 405-416.

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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 
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Shang, Jen-Ling. Branch-Weight Unique Trees. Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 2, pp. 405-416. http://geodesic.mathdoc.fr/item/DMGT_2022_42_2_a5/

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