For a connected graph G and a, b is an element of R, the general degree -eccentricity index is defined as DEIa,b(G) = Sigma(v is an element of V(G)) d(G)(a)(v)ecc(G)(b)(v), where V(G) is the vertex set of G, d(G)(v) is the degree of a vertex v and ecc(G)(v) is the eccentricity of v in G, i.e. the maximum distance from v to another vertex of the graph. This index generalizes several well-known 'topological indices' of graphs such as the eccentric connectivity index. We characterize the unique trees with the maximum and the minimum general degree-eccentricity index among all n -vertex trees with fixed maximum degree for the cases a >= 1, b <= 0 and 0 <= a <= 1, b >= 0. This complements previous results on the general degree -eccentricity index for various classes of trees.