  We define a notion of stochastic domination between trees, where one tree
dominates another if when the vertices of each are labeled with independent,
identically distributed random variables, one tree is always more likely to
contain a path with a specified property. Sufficient conditions for this kind
of domination are (1) more symmetry and (2) earlier branching. We apply these
conditions to the problem of determining how fast a tree must grow before
first-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 each
vertex at distance n-1 from the root has f(n) offspring, f nondecreasing, an
explosion occurs with exponentially distributed passage times if and only if
sum f(n)^{-1}<infty.
