  Patterns on numerical semigroups are multivariate linear polynomials, and
they are said to be admissible if there exists a numerical semigroup such that
evaluated at any nonincreasing sequence of elements of the semigroup gives
integers belonging to the semigroup. In a first approach, only homogeneous
patterns where analized. In this contribution we study conditions for an
eventually non-homogeneous pattern to be admissible, and particularize this
study to the case the independent term of the pattern is a multiple of the
multiplicity of the semigroup. Moreover, for the so called strongly admissible
patterns, the set of numerical semigroups admitting these patterns with fixed
multiplicity $m$ form an $m$-variety, which allows us to represent this set in
a tree and to describe minimal sets of generators of the semigroups in the
variety with respect to the pattern. Furthermore, we characterize strongly
admissible patterns having a finite associated tree.
