The eccentricity matrix epsilon(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. In this paper we focus on the eccentricity matrix of graphs. Let T be an n-vertex tree and let epsilon(n)(T) be the least epsilon-eigenvalue of T. On the one hand, we determine the n-vertex trees with the minimum epsilon-spectral radius. On the other hand, for n ...