  The eccentricity matrix of a connected graph $G$, denoted by
$\mathcal{E}(G)$, is obtained from the distance matrix of $G$ by keeping the
largest nonzero entries in each row and each column and leaving zeros in the
remaining ones. The eigenvalues of $\mathcal{E}(G)$ are the
$\mathcal{E}$-eigenvalues of $G$. The eccentricity energy (or the
$\mathcal{E}$-energy) of $G$ is the sum of the absolute values of all
$\mathcal{E}$-eigenvalues of $G$. In this article, we determine the unique tree
with the minimum second largest $\mathcal{E}$-eigenvalue among all trees on $n$
vertices other than the star. Also, we characterize the trees with minimum
$\mathcal{E}$-energy among all trees on $n$ vertices.
