  We consider the eccentric graph of a graph $G$, denoted by $ecc(G)$, which
has the same vertex set as $G$, and two vertices in the eccentric graph are
adjacent iff their distance in $G$ is equal to the eccentricity of one of them.
In this paper, we present a fundamental requirement for the isomorphism between
$ecc(G)$ and the complement of $G$, and show that the previous necessary
condition given in the literature is inadequate. Also we obtain that diameter
of $ecc(T)$ is at most $3$ for any tree and get some characterizations of the
eccentric graph of trees.
