  Let $p',q'\in R^n$. Write $p'\sim q'$ if $p'-q'$ is a multiple of
$(1,\ldots,1)$. Two different points $p$ and $q$ in $R^n/\sim$ 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 $\dd(p,pq)$, $\dd(qp,q)$ and $\dd(p,q)$ 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 $A$.
