Distributive lattices of tilting modules and support $\tau$-tilting modules over path algebras

In this paper we study the poset of basic tilting $kQ$-modules when $Q$ is a Dynkin quiver, and the poset of basic support $\tau$-tilting $kQ$-modules when $Q$ is a connected acyclic quiver respectively. It is shown that the first poset is a distributive lattice if and only if $Q$ is of types $$\mathbb{A}${1}, $\mathbb{A}${2}$or$$\mathbb{A}$_{3}$ with a nonlinear orientation and the second poset is a distributive lattice if and only if $Q$ is of type ${\mathbb{A}}_1$.