  The contact process on an infinite homogeneous tree is shown to exhibit at
least two phase transitions as the infection parameter lambda is varied. For
small values of lambda a single infection eventually dies out. For larger
lambda the infection lives forever with positive probability but eventually
leaves any finite set. (The survival probability is a continuous function of
lambda, and the proof of this is much easier than it is for the contact process
on d-dimensional integer lattices.) For still larger lambda the infection
converges in distribution to a nontrivial invariant measure. For an n-ary tree,
with n large, the first of these transitions occurs when lambda~1/n and the
second occurs when 1/2 sqrt{n}<lambda<e/sqrt{n}. Nonhomogeneous trees whose
vertices have degrees varying between 1 and n behave essentially as homogeneous
n-ary trees, provided that vertices of degree n are not too rare. In
particular, letting n go to infty, Galton-Watson trees whose vertices have
degree n with probability that does not decrease exponentially with n may have
both phase transitions occur together at lambda=0. The nature of the second
phase transition is not yet clear and several problems are mentioned in this
regard.
