  We establish compact presentability, i.e. the locally compact version of
finite presentability, for an infinite family of tree almost automorphism
groups. Examples covered by our results include Neretin's group of
spheromorphisms, as well as the topologically simple group containing the
profinite completion of the Grigorchuk group constructed by Barnea, Ershov and
Weigel.
  We additionally obtain an upper bound on the Dehn function of these groups in
terms of the Dehn function of an embedded Higman-Thompson group. This, combined
with a result of Guba, implies that the Dehn function of the Neretin group of
the regular trivalent tree is polynomially bounded.
