  We study the extinction time $\uptau$ of the contact process on finite trees
of bounded degree. We show that, if the infection rate is larger than the
critical rate for the contact process on $\Z$, then, uniformly over all trees
of degree bounded by a given number, the expectation of $\uptau$ grows
exponentially with the number of vertices. Additionally, for any sequence of
growing trees of bounded degree, $\uptau$ divided by its expectation converges
in distribution to the unitary exponential distribution. These also hold if one
considers a sequence of graphs having spanning trees with uniformly bounded
degree. Using these results, we consider the contact process on a random graph
with vertex degrees following a power law. Improving a result of Chatterjee and
Durrett \cite{CD}, we show that, for any infection rate, the extinction time
for the contact process on this graph grows exponentially with the number of
vertices.
