# Branch duplication in trees

## Authors

• Charles R. Johnson The College of William and Mary.
• Jacob Lettie Duke University.
• Sander Mack-Crane Case Western Reserve University.
• Alicja Szabelska-Beręsewicz Poznan University of Life Sciences.

## Keywords:

Branch duplication in trees, Enumeration of seeds, Rooted trees, Seeds, Uniqueness of seeds

## Abstract

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.

## Author Biographies

### Charles R. Johnson, The College of William and Mary.

Department of Mathematics,

### Jacob Lettie, Duke University.

Department of Mathematics.

### Sander Mack-Crane, Case Western Reserve University.

Department of Mathematics, Applied Mathematics and Statistics.

### Alicja Szabelska-Beręsewicz, Poznan University of Life Sciences.

Dept. of Mathematical and Statistical Methods.

## References

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.

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

A. Leal-Duarte and C. 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

The R Fundation, “The R Project for Statistical Computing”, R-project, 2016. [Online]. Available: https://www.r-project.org/.

2020-04-29

## How to Cite

[1]
C. R. Johnson, J. Lettie, S. Mack-Crane, and A. Szabelska-Beręsewicz, “Branch duplication in trees: uniqueness of seeds and enumeration of seeds”, Proyecciones (Antofagasta, On line), vol. 39, no. 2, pp. 451-465, Apr. 2020.

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