  We prove bubble-tree convergence of sequences of gradient Ricci shrinkers
with uniformly bounded entropy and uniform local energy bounds, refining the
compactness theory of Haslhofer-Mueller. In particular, we show that no energy
concentrates in neck regions, a result which implies a local energy identity
for the sequence. Direct consequences of these results are an identity for the
Euler characteristic and a local diffeomorphism finiteness theorem.
