  The eccentricity matrix $\varepsilon(G)$ of a graph $G$ is constructed from
the distance matrix of $G$ by keeping only the largest distances for each row
and each column. This matrix can be interpreted as the opposite of the
adjacency matrix obtained from the distance matrix by keeping only the
distances equal to 1 for each row and each column. The
$\varepsilon$-eigenvalues of a graph $G$ are those of its eccentricity matrix
$\varepsilon(G)$. Wang et al \cite{e} proposed the problem of determining the
maximum $\varepsilon$-spectral radius of trees with given order. In this paper,
we consider the above problem of $n$-vertex trees with given diameter. The
maximum $\varepsilon$-spectral radius of $n$-vertex trees with fixed odd
diameter is obtained, and the corresponding extremal trees are also determined.
The trees with least $\varepsilon$-eigenvalues in $[-2\sqrt{2},0)$ have been
known. Finally, we determine the trees with least $\varepsilon$-eigenvalues in
$[-2-\sqrt{13},-2\sqrt{2})$.
