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arXiv · 2605.03681

Magnitude and diversity of trees

Philippe Bouafia
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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.

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Philippe Bouafia. 2026-05-05. Magnitude and diversity of trees. https://arxiv.org/abs/2605.03681

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