Self-embeddings of computable trees
  We divide the class of infinite computable trees into three types. For the first and second types, $0'$ computes a nontrivial self-embedding while for the third type $0''$ computes a nontrivial self-embedding. These results are optimal and we obtain partial results concerning the complexity of nontrivial self-embeddings of infinite computable trees considered up to isomorphism. We show that every infinite computable tree must have either an infinite computable chain or an infinite $\Pi^0_1$ antichain. This result is optimal and has connections to the program of reverse mathematics. 
Stephen Binns, Bj{\o}rn Kjos-Hanssen, Manuel Lerman, James H. Schmerl,   and Reed Solomon
math.LO