  We consider the contact process with infection rate $\lambda$ on
$\mathbb{T}_n^d$, the $d$-ary tree of height $n$. We study the extinction time
$\tau_{\mathbb{T}_n^d}$, that is, the random time it takes for the infection to
disappear when the process is started from full occupancy. We prove two
conjectures of Stacey regarding $\tau_{\mathbb{T}_n^d}$. Let $\lambda_2$ denote
the upper critical value for the contact process on the infinite $d$-ary tree.
First, if $\lambda < \lambda_2$, then $\tau_{\mathbb{T}_n^d}$ divided by the
height of the tree converges in probability, as $n \to \infty$, to a positive
constant. Second, if $\lambda > \lambda_2$, then $\log
\mathbb{E}[\tau_{\mathbb{T}_n^d}]$ divided by the volume of the tree converges
in probability to a positive constant, and
$\tau_{\mathbb{T}_n^d}/\mathbb{E}[\tau_{\mathbb{T}_n^d}]$ converges in
distribution to the exponential distribution of mean 1.
