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We define a notion of stochastic domination between trees, where one treedominates another if when the vertices of each are labeled with independent,identically distributed random variables, one tree is always more likely tocontain a path with a specified property. Sufficient conditions for this kindof domination are (1) more symmetry and (2) earlier branching. We apply theseconditions to the problem of determining how fast a tree must grow beforefirst-passage percolation on the tree exhibits an explosion, that is to say,infinitely many vertices are reached in finite time. For a tree in which eachvertex at distance n-1 from the root has f(n) offspring, f nondecreasing, anexplosion occurs with exponentially distributed passage times if and only ifsum f(n)^{-1}<infty.

Domination Between Trees and Application to an Explosion Problem