On the isometry groups of certain CAT(0) spaces and trees

We show that the automorphism group of a locally finite tree is discrete, or pro-finite, or not the inverse limit of an inverse system of discrete groups, and provide necessary and sufficient conditions for each of these possibilities to occur. More generally, we demonstrate that for certain proper CAT(0) spaces X, the group of isometries of X is not an inverse limit of Lie groups.