Length-minimizing LED Trees

TL;DR

This paper introduces length-minimizing LED trees, a special Euclidean tree structure relevant to phylogeny, and proves their uniqueness along with geometric properties, with an application in historical linguistics.

Contribution

It provides a uniqueness proof for length-minimizing LED trees and explores their geometric and topological properties, extending Steiner tree concepts.

Findings

Proved the uniqueness of length-minimizing LED trees.

Explored geometric and topological properties of the feasible set.

Demonstrated an application in historical linguistics.

Abstract

In this paper, we introduce a specific type of Euclidean tree called LED (Leaves of Equal Depth) tree. LED trees can be used in computational phylogeny, since they are a natural representative of the time evolution of a set of species in a feature space. This work is focused on LED trees that are length minimizers for a given set of leaves (species) and a given isomorphism type (the hierarchical structure of ancestors). The underlying minimization problem can be seen as a variant of the classical Euclidean Steiner tree problem. Even though it has a convex objective function, it is rather non-trivial, since it has a non-convex feasible set. The main contribution of this paper is that we provide a uniqueness result for this problem. Moreover, we explore some geometrical and topological properties of the feasible set and we prove several geometrical characteristics of the length minimizers that are analogical to the properties of Steiner trees. At the end, we show a simple example of an application in historical linguistics.