  The aim of this paper is to study a poset isomorphism between two support
$\tau$-tilting posets. We take several algebraic information from combinatorial
properties of support $\tau$-tilting posets. As an application, we treat a
certain class of basic algebras which contains preprojective algebras of type
$A$, Nakayama algebras, and generalized Brauer tree algebras. We provide a
necessary condition for that an algebra $\Lambda$ share the same support
$\tau$-tilting poset with a given algebra $\Gamma$ in this class. Furthermore,
we see that this necessary condition is also a sufficient condition if $\Gamma$
is either a preprojective algebra of type $A$, a Nakayama algebra, or a
generalized Brauer tree algebra.
