There are certainly connections. König's Lemma isn't constructively true, for example. That is, given a computable description of an infinite tree, the problem of finding an infinite path may not be computable. Imagine walking down the tree node by node -- how can you tell whether you have stepped into a merely finite but enormously large subtree and have missed the infinite branch? Obviously, an oracle for the halting problem will solve that problem. I don't know off-hand, or from thinking about it for five minutes, whether the reverse is the case, i.e. if the halting problem can be encoded as the problem of finding an infinite branch in some computable infinite tree.