2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/134791We study the relation between the minimal spanning tree (MST) on many random points and the "near-minimal" tree which is optimal subject to the constraint that a proportion $δ$ of its edges must be different from those of the MST. Heuristics suggest that, regardless of details of the probability model, the ratio of lengths should scale as $1 + Θ(δ^2)$. We prove this scaling result in the model of the lattice with random edge-lengths and in the Euclidean model.24 pages, 3 figuresProbability05C80; 60K35; 68W40text