  Evans defines a notion of what it means for a set B to be polar for a process
indexed by a tree. The main result herein is that a tree picked from a
Galton-Watson measure whose offspring distribution has mean m and finite
variance will almost surely have precisely the same polar sets as a
deterministic tree of the same growth rate. This implies that deterministic and
nondeterministic trees behave identically in a variety of probability models.
Mapping subsets of Euclidean space to trees and polar sets to capacity
criteria, it follows that certain random Cantor sets are capacity-equivalent to
each other and to deterministic Cantor sets. An extension to branching
processes in varying environment is also obtained.
