This is the second talk of my dissertation examination. For every simple closed curve of a surface, its dual tree is a tree embedded in the universal cover of a surface, each edge of which is associated with a liftof the curve. It gives information about the curve, for example, the intersection number with another simple closed curve. We will focus on the properties of dual trees and the proof of the main theorem which informs right-angled Artin subgroups of mappingclass groups.