Magnitude and diversity of trees

TL;DR

This paper calculates the magnitude of compact R-trees, revealing it equals 1 plus half their total length, and explores how diversity measures relate to tree structure and computational complexity.

Contribution

It establishes a formula for the magnitude of R-trees and analyzes the sensitivity of diversity-maximizing measures to branching structures.

Findings

Magnitude of compact R-trees equals 1 + L/2, where L is total length.

Diversity-maximizing measures tend to concentrate on leaves, avoiding branch points.

Maximum diversity on weighted trees can be computed in polynomial time.

Abstract

We compute the magnitude (an isometric invariant of metric spaces) of compact $\mathbb{R}$-trees and show that it equals $1 + L/2$, where $L \in [0, \infty]$ denotes the total length. Although length is the only geometric invariant captured by magnitude, we show that diversity-maximizing measures on compact $\mathbb{R}$-trees are more sensitive to the branching structure as they tend to be more concentrated toward the leaves: their support contains no branch points. In the finite case, we further show that maximum diversity on a weighted tree can be computed in polynomial time.