Looking at this in broader context, the Christological implications are fascinating.
Absalom was David's son. Guess who else was the "Son of David?" Christ. After Absalom's death, Israel was restored to order. His death brought about peace for Israel, just like Christ's death brings peace (in heaven) to God's people.
Absalom died on a tree. So did Christ. Absalom was finished off by a spear. So was Christ.
I think this is a sort of reverse metaphor, though. (I'm sure there's better terminology for that.) Unlike Christ, Absalom was a prideful, arrogant, sinful, rebellious jerk. Christ was a humble servant. Absalom undermined his father, Christ went along with his Father's plan despite the pain he knew he'd have to endure. Absalom was fleeing the battle, Jesus humbly and willingly approached his death.
As to David's role, the opening verses of 2 Samuel 9 are quite interesting. Joab rebukes David in verse 6, saying
for today I know that if Absalom were alive and all of us were dead
today, then you would be pleased.
David wished he could sacrifice his innocent people for his sinful son. Remind you of anything? God sacrificed his innocent Son for his sinful people!
We say the Old Testament points to Christ alot, and I think it does. However, often it's by reverse metaphor. David portrayed God the Father here, but he did a pretty crappy job. Absalom portrayed Christ, but he screwed up too.
Thankfully, God's plans are a lot better and the reality is greater than the promise.