  A little over 25 years ago Pemantle pioneered the study of the contact
process on trees, and showed that on homogeneous trees the critical values
$\lambda_1$ and $\lambda_2$ for global and local survival were different. He
also considered trees with periodic degree sequences, and Galton-Watson trees.
Here, we will consider periodic trees in which the number of children in
successive generation is $(n,a_1,\ldots, a_k)$ with $\max_i a_i \le
Cn^{1-\delta}$ and $\log(a_1 \cdots a_k)/\log n \to b$ as $n\to\infty$. We show
that the critical value for local survival is asymptotically $\sqrt{c (\log
n)/n}$ where $c=(k-b)/2$. This supports Pemantle's claim that the critical
value is largely determined by the maximum degree, but it also shows that the
smaller degrees can make a significant contribution to the answer.
