Consider the contact process on a homogeneous tree with degree $d
\geq 3$. Denote by
$$\lambda_c = \inf\{\lambda : P(o \in \xi^o_t
\text{i.o.}) > 0\}$$
the critical value of local survival probability,
where o is the root of the tree. Pemantle and Durrett and Schinazi both
conjectured that the complete convergence theorem should hold if $\lambda
>\lambda_c$. Here we answer the conjecture affirmatively. Furthermore, we
will show that
$$P(o \in \xi^o_t \text{i.o.}) = 0 \quad \text{at
$\lambda_c}.$$
Therefore, the conclusion of the complete convergence
theorem cannot hold at $\lambda_c$