  We study isometric actions of tree automorphism groups on the
infinite-dimensional hyperbolic spaces. On the one hand, we exhibit a general
one-parameter family of such representations and analyse the corresponding
equivariant embeddings of the trees, showing that they are convex-cocompact and
asymptotically isometric. On the other hand, focusing on the case of
sufficiently transitive groups of automorphisms of locally finite trees, we
classify completely all irreducible representations by isometries of hyperbolic
spaces. It turns out that in this case our one-parameter family exhausts all
non-elementary representations.
