  An order-theoretic forest is a countable partial order such that the set of
elements larger than any element is linearly ordered. It is an order-theoretic
tree if any two elements have an upper-bound. The order type of a branch can be
any countable linear order. Such generalized infinite trees yield convenient
definitions of the rank-width and the modular decomposition of countable
graphs.
  We define an algebra based on only four operations that generate up to
isomorphism and via infinite terms these order-theoretic trees and forests. We
prove that the associated regular objects, those defined by regular terms, are
exactly the ones that are the unique models of monadic second-order sentences.
