  For a reduced one-dimensional complete local $k$-algebra $R$, Huneke et al.
(Res. Math. Sci., 8(4), paper no. 60, 2021) introduced an important invariant,
the reduced type. In this article, we study the extremal behavior of reduced
type of some special numerical semigroup rings. For a numerical semigroup ring,
the behavior of reduced type can be studied by analyzing the set of
pseudo-Frobenius elements of the numerical semigroup. We give complete
descriptions of pseudo-Frobenius elements of Bresinsky's numerical semigroups
and duplication of numerical semigroups. Further, we explore the extremal
behavior of reduced type for the associated semigroup rings.
