Magnitude and diversity of trees
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866914531075162112 |
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| author | Bouafia, Philippe |
| author_facet | Bouafia, Philippe |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_03681 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Magnitude and diversity of trees Bouafia, Philippe Metric Geometry 51F99, 30L99, 05C05 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. |
| title | Magnitude and diversity of trees |
| topic | Metric Geometry 51F99, 30L99, 05C05 |
| url | https://arxiv.org/abs/2605.03681 |