  We consider random walks indexed by arbitrary finite random or deterministic
trees. We derive a simple sufficient criterion which ensures that the maximal
displacement of the tree-indexed random walk is determined by a single large
jump. This criterion is given in terms of four quantities : the tail and the
expectation of the random walk steps, the height of the tree and the number of
its vertices. The results are applied to critical Galton--Watson trees with
offspring distributions in the domain of attraction of a stable law.
