  We study 2-term tilting complexes of Brauer tree algebras in terms of
simplicial complexes. We show the symmetry and convexity of the simplicial
complexes as lattice polytopes. Via a geometric interpretation of derived
equivalences, we show that the $f$-vector of simplicial complexes of Brauer
tree algebras only depends the number of the edges of the Brauer trees and
hence it is a derived invariant. In particular, this result implies that the
number of 2-term tilting complexes, which is in bijection with support
$\tau$-tilting modules, is a derived invariant. Moreover, we apply our result
to the enumeration problem of Coxeter-biCatalan combinatorics.
