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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 todisappear when the process is started from full occupancy. We prove twoconjectures of Stacey regarding $\tau_{\mathbb{T}_n^d}$. Let $\lambda_2$ denotethe 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 theheight of the tree converges in probability, as $n \to \infty$, to a positiveconstant. Second, if $\lambda > \lambda_2$, then $\log\mathbb{E}[\tau_{\mathbb{T}_n^d}]$ divided by the volume of the tree convergesin probability to a positive constant, and$\tau_{\mathbb{T}_n^d}/\mathbb{E}[\tau_{\mathbb{T}_n^d}]$ converges indistribution to the exponential distribution of mean 1.

The contact process on finite homogeneous trees revisited