The Length of a Minimal Tree With a Given Topology: generalization of Maxwell Formula

TL;DR

This paper generalizes Maxwell's formula to compute the length of arbitrary minimal trees with a given topology in any Euclidean space, eliminating the need to identify degenerate edges or their directions.

Contribution

The authors extend Maxwell's formula to arbitrary extreme trees in any Euclidean space, simplifying length calculations without requiring edge degeneracy information.

Findings

Generalized Maxwell formula for arbitrary trees

Applicable in Euclidean spaces of any dimension

Length computed as maximum of a linear function over a convex set

Abstract

The classic Maxwell formula calculates the length of a planar locally minimal binary tree in terms of coordinates of its boundary vertices and directions of incoming edges. However, if an extreme tree with a given topology and a boundary has degenerate edges, then the classic Maxwell formula cannot be applied directly, to calculate the length of the extreme tree in this case it is necessary to know which edges are degenerate. In this paper we generalize the Maxwell formula to arbitrary extreme trees in a Euclidean space of arbitrary dimension. Now to calculate the length of such a tree, there is no need to know either what edges are degenerate, or the directions of nondegenerate boundary edges. The answer is the maximum of some special linear function on the corresponding compact convex subset of the Euclidean space coinciding with the intersection of some cylinders.