  Inspired by the reconstruction program of conformal field theories of Vaughan
Jones we recently introduced a vast class of so called forest-skein groups.
They are built from a skein presentation: a set of colours and a set of pairs
of coloured trees. Each nice skein presentation produces four groups similar to
Richard Thompson's group F,T,V and the braided version BV of Brin and Dehornoy.
  In this article, we consider forest-skein groups obtained from
one-dimensional skein presentations; the data of a homogeneous monoid
presentation. We decompose these groups as wreath products. This permits to
classify them up to isomorphisms. Moreover, we prove that a number of
properties of the fraction group of the monoid pass through the forest-skein
groups such as the Haagerup property, homological and topological finiteness
properties, and orderability.
