  We determine bubble tree convergence for a sequence of harmonic maps, with
uniform energy bounds, from a compact Riemann surface into a compact locally
CAT(1) space. In particular, we demonstrate energy quantization and the no-neck
property for such a sequence. In the smooth setting, Jost and Parker
respectively established these results by exploiting now classical arguments
for harmonic maps. Our work demonstrates that these results can be
reinterpreted geometrically. In the absence of a PDE, we take advantage of the
local convexity properties of the target space. Included in this paper are an
$\epsilon$-regularity theorem, an energy gap theorem, and a removable
singularity theorem for harmonic maps for harmonic maps into metric spaces with
upper curvature bounds. We also prove an isoperimetric inequality for conformal
harmonic maps with small image.
