The true Duke—Duke Senior—has taken refuge in the forest in the way that the lines of the song sung under the greenwood tree suggest that people should.
When Duke Frederick usurps his older brother's position, Duke Senior runs away to the forest. Most of the characters are not aware that he is there until they, too, flee Duke Frederick and move to the Forest of Arden. In the forest, he finds peace and relaxation. He finds refuge from his brother's machinations.
Amiens sings a song about lying under the greenwood tree, relaxing and only worrying about the possibility of bad weather. The song offers a place of refuge from the troubles of the world by choosing to live in and focus on nature. It says to shun ambition and instead live off the land and provide for yourself.
This is essentially what Duke Senior has done in the Forest of Arden. He has removed himself from his major enemy—his brother—and found a place of refuge.
Jacques adds a verse to the song that says when people shun wealth and ease only to please their stubborn will, they're fools. He doesn't seem to share the view of Amiens that the forest is a completely positive thing for people like Duke Senior and the other nobles who have run from Duke Frederick.