  In this work we introduce the notion of almost-symmetry for generalized
numerical semigroups. In addition to the main properties occurring in this new
class, we present several characterizations for its elements. In particular we
show that this class yields a new family of Frobenius generalized numerical
semigroups and extends the class of irreducible generalized numerical
semigroups. This investigation allows us to provide a method of computing all
almost symmetric generalized numerical semigroup having a fixed Frobenius
element and organizing them in a rooted tree depending on a chosen monomial
order.
