  The \textit{eccentricity matrix} $\mathcal{E}(G)$ of a connected graph $G$ is
obtained from the distance matrix of $G$ by keeping the largest non-zero
entries in each row and each column, and leaving zeros in the remaining ones.
The eigenvalues of $\mathcal{E}(G)$ are the \textit{$\mathcal{E}$-eigenvalues}
of $G$. In this article, we find the inertia of the eccentricity matrices of
trees. Interestingly, any tree on more than $4$ vertices with odd diameter has
two positive and two negative $\mathcal{E}$-eigenvalues (irrespective of the
structure of the tree). A tree with even diameter has the same number of
positive and negative $\mathcal{E}$-eigenvalues, which is equal to the number
of 'diametrically distinguished' vertices (see Definition 3.1). Besides we
prove that the spectrum of the eccentricity matrix of a tree is symmetric with
respect to the origin if and only if the tree has odd diameter. As an
application, we characterize the trees with three distinct
$\mathcal{E}$-eigenvalues.
