The contact process on $Z^d$ is known to have only two fundamental
types of behavior: survival and extinction. Recently Pemantle discovered that
the phase structure for the contact process on a tree can be more complex.
There are three possible types of behavior: strong survival, weak survival and
extinction. He proved that all three occur on homogeneous trees in which each
vertex has $d + 1$ neighbors, provided that $d \geq 3$, but he left open the
case $d = 2$. Since $d = 1$ corresponds to $Z^1$, in which weak survival does
not occur, $d = 2$ is the boundary case. In this paper, we complete this
picture, by showing that weak survival does occur on the binary tree for
appropriate parameter values. In doing so, we extend and develop techniques for
obtaining upper and lower bounds for the critical values associated with strong
and weak survival of the contact process on more general graphs.