Some upper bounds for minimal trees

Abstract

This paper introduces two upper bounds for the length of two kinds of minimal trees. The first upper bound, σ(n) < π ([((n − 1) π) 1 2 ] + 1 2 )+ 1 2 , is related to the Euclidean Steiner Minimal Tree obtained from any set of n points enclosed i nside a circumference of radius 1. The second upper bound, Φ(n)⩽2(n − 1) k , is related to the Rectilinear Minimal Tree obtained from any set of n = [(k(k+2)+1) 2] points ( k=1, 2, 3,…),, enclosed in a square with sides equal to 1. Moreover, we also present the only case where the well-known bound σ(n) = 1 + n is attained for the length of the Rectilinear Steiner Minimal Tree for any set of n = t 2 points ,( t=2, 3,…), enclosed in a square with sides equal to 1.