  In this paper, we consider the threshold-one contact process and the
threshold-one voter model w/o spontaneous death on homogeneous trees
$\mathbb{T}_d$, $d\ge 2$. Mainly inspired by the corresponding arguments for
ordinary contact processes, we prove that the complete convergence theorem
holds for these three systems under strong survival. When the systems survives
weakly, complete convergence may also hold under certain transition and/or
initial conditions.
