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Proyecciones (Antofagasta)

versión impresa ISSN 0716-0917

Proyecciones (Antofagasta) vol.39 no.2 Antofagasta abr. 2020 


Branch duplication in trees: uniqueness of sedes and enumeration of sedes

Charles R. Johnson1 

Jacob Lettie2 

Sander Mack-Crane3 

Alicja Szabelska-Beręsewicz4

1The College of William and Mary, Dept. of Mathematics, Williamsburg, VA, U.S.A.: e-mail:

2Duke University, Dept.of Mathematics, Durham, NC, U.S.A. e-mail:

3Case Western Reserve University, Dept. of Mathematics, Applied Mathematics and Statistics, Cleveland, OH, U.S.A. e-mail:

4Poznan University of Life Sciences, Dept. of Mathematical and Statistical Methods, Poznan, Poland. e-mail:


By the process of branch duplication, any tree may be generated from a type of tree called a seed. We describe the correspondence between trees and seeds by showing that each tree grows from a unique seed and giving an algorithm to produce this seed. Using results about the structure of seeds, we enumerate the seeds of a given diameter.

Keywords: Branch duplication in tres; Enumeration of sedes; Rooted tres; Seeds; Uniqueness of seeds

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This work was supported by NSF grant DMS 075 1964.


[1] C. R. Johnson and C. M. Saiago, “Branch duplication for the construction of multiple eigenvalues in an Hermitian matrix whose graph is a tree”,Linear and multilinear algebra, vol. 56, no. 4, pp. 357-380, 2008, doi: 10.1080/03081080600597668. [ Links ]

[2] C. R. Johnson andC. M. Saiago , “Diameter minimal trees”, Linear and multilinear algebra, vol. 64, no. 3, pp. 557-571, 2015, doi: 10.1080/03081087.2015.1057097. [ Links ]

[3] A. Leal-Duarte andC. R. Johnson , “On the minimum number of distinct eigenvalues for a symmetric matrix whose graph is a given tree”,Mathematical Inequalities & Applications, vol. 5, no. 2, pp. 175-180, 2002, doi: 10.7153/mia-05-19. [ Links ]

[4] The R Fundation, “The R Project for Statistical Computing”, R-project, 2016. [Online]. Available: [ Links ]

Received: June 30, 2019; Accepted: November 30, 2019

Creative Commons License This is an open-access article distributed under the terms of the Creative Commons Attribution License