  Parabolic (resp. hyperbolic) self-embeddings of trees are those which do not
fix a non-empty finite subtree and preserve precisely one (resp. two) end(s).
We prove that a locally finite tree having a parabolic self-embedding is
mutually embeddable with infinitely many pairwise non-isomorphic trees, unless
the tree is a one-way infinite path. As a result, we conclude that two
important properties identified by Bonato-Tardif and Tyomkyn hold for locally
finite trees not having any hyperbolic self-embedding.
