How does Absalom's death foreshadow Christ?

Absalom dies on a tree (2 Samuel 18). He's rebelled against his father, David, and taken the throne. He'd rallied all Israel to his side. David has fled, but he's sent in his army and they've totally crushed Absalom. Absalom is fleeing into the forest when he gets caught, hangs on the tree, and eventually gets speared to death by Joab (the leader of David's army). Reminds me of Jesus's death on a cross. After hearing of Absalom's death, David is sad. *really sad*. So sad he can't even rejoice that he's gotten his kingdom back. Is there any Christological significance to this scenario? What about David? Does he play a metaphorical role?