  We produce two separate algebraic descriptions of the isomorphism classes of
the solvable subgroups of the group PLo(I) of piecewise-linear
orientation-preserving homeomorphisms of the unit interval under the operation
of composition, and also of the solvable subgroups of the generalized R.
Thompson groups F_n. The first description is as a set of isomorphism classes
of groups which is closed under three algebraic operations, and the second is
as the set of isomorphism classes of subgroups of a countable collection of
easily described groups. We show the two descriptions are equivalent.
