  We show that any Brauer tree algebra has precisely $\binom{2n}{n}$
$2$-tilting complexes, where $n$ is the number of edges of the associated
Brauer tree. More explicitly, for an external edge $e$ and an integer $j\neq0$,
we show that the number of $2$-tilting complexes $T$ with $g_e(T)=j$ is
$\binom{2n-|j|-1}{n-1}$, where $g_e(T)$ denotes the $e$-th of the $g$-vector of
$T$. To prove this, we use a geometric model of Brauer graph algebras on the
closed oriented marked surfaces and a classification of $2$-tilting complexes
due to Adachi-Aihara-Chan.
