2026/05/05 by Philippe Bouafia · 1 voice
Mathematics · #math.MG
We compute the magnitude (an isometric invariant of metric spaces) of compact ℝ-trees and show that it equals 1 + L/2, where L ∈ [0, ∞] denotes the total length. Although length is the only geometric invariant captured by magnitude, we show that diversity-maximizing measures on compact 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.