Hawke likes to explore. It's kind of his thing. If something catches his interest that's not on the map Hawke will definitely want to check it out. Honestly he would mostly just ignore the map and encourage the group to explore as well.

If they still end up at the tree, Hawke's first instinct would be to stab the heart. A black, human heart sitting in the center of an old tree just screams bad news to him. He wouldn't trust anything about that tree at all.