Distances on the tropical line determined by two points

Let p′ and q′ be points in R. Write p′ ∼ q′ if p′ − q′ is a multiple of (1, . . . , 1). Two different points p and q in R/ ∼ uniquely determine a tropical line L(p, q) passing through them and stable under small perturbations. This line is a balanced unrooted semi–labeled tree on n leaves. It is also a metric graph. If some representatives p′ and q′ of p and q are the first and second columns of some real normal idempotent order n matrix A, we prove that the tree L(p, q) is described by a matrix F , easily obtained from A. We also prove that L(p, q) is caterpillar. We prove that every vertex in L(p, q) belongs to the tropical linear segment joining p and q. A vertex, denoted pq, closest (w.r.t tropical distance) to p exists in L(p, q). Same for q. The distances between pairs of adjacent vertices in L(p, q) and the distances d(p, pq), d(qp, q) and d(p, q) are certain entries of the matrix |F |. In addition, if p and q are generic, then the tree L(p, q) is trivalent. The entries of F are differences (i. e., sum of principal diagonal minus sum of secondary diagonal) of order 2 minors of the first two columns of A.