  We consider the simple random walk on Galton-Watson trees with supercritical
offspring distribution, conditioned on non-extinction. In case the offspring
distribution has finite support, we prove an upper bound for the annealed
return probability to the root which decays subexponentially in time with
exponent 1/3. This exponent is optimal. Our result improves the previously
known subexponential upper bound with exponent 1/5 by Piau [Ann. Probab. 26,
1016-1040 (1998)]. For offspring distributions with unbounded support but
sufficiently fast decay, our method also yields improved subexponential upper
bounds.
