  We say that a real X is n-generic relative to a perfect tree T if X is a path
through T and for all Sigma^0_n (T) sets S, there exists a number k such that
either X|k is in S or for all tau in T extending X|k we have tau is not in S. A
real X is n-generic relative to some perfect tree if there exists such a T. We
first show that for every number n all but countably many reals are n-generic
relative to some perfect tree. Second, we show that proving this statement
requires ZFC^- + ``There exist infinitely many iterates of the power set of the
natural numbers''. Third, we prove that every finite iterate of the hyperjump,
O^(n), is not 2-generic relative to any perfect tree and for every ordinal
alpha below the least lambda such that sup_{beta < lambda} (beta th admissible)
= lambda, the iterated hyperjump O^(alpha) is not 5-generic relative to any
perfect tree. Finally, we demonstrate some necessary conditions for reals to be
1-generic relative to some perfect tree.
