On the number of automorphism of uncontable models

Let s(A) denote the number of automorphisms of a model A of power omega_1. We derive a necessary and sufficient condition in terms of trees for the existence of an A with omega_1 < s(A) < 2^{omega_1}. We study the sufficiency of some conditions for s(A)=2^{omega_1}. These conditions are analogous to conditions studied by D.Kueker in connection with countable models.