Mapping class groups of finite type surfaces and right-angled Artin groups are well-studied in low-dimensional topology and geometric group theory. By Nielsen--Thurston classification, every mapping class is either periodic, reducible or pseudo-Anosov. Koberda showed that (in soft situation) sufficiently large powers of Dehn twists and purely pseudo-Anosov mapping classes (on subsurfaces) generate a right-angled Artin subgroup of a mapping class group. In this talk, we will deal with tree actions of surface groups and calculate a lower bound of exponents for powers of Dehn twists to generate a right-angled Artin subgroup.