Embed my sword in the tree - I don't deserve to be using a weapon of honor anymore.
Dishonored by the fact that you can no longer serve your God, you essentially embrace the path of a Righteous assassin, and you embed your sword to a tree. Thinking that that was a wholly foolish idea, you instead bury it in the ground and mark the ground where you buried it. You tighten the buckles of your hand guard, and now, armed with a new way of dealing with problems, you head back to the cave, and you grab the dark sword which Kolkoli has prepared for you rather nicely. "John was right, there will be a reckoning for all of this. This sword represents my tainted honor that I must live with!" with that, you sheath your new beautiful, sword.
YOU LOSE: Iron Highlander-Esque Sword
YOU GAIN: The Dishonorable Edge
(Boom, I just turned you in to batman who actually kills people.)
No need to hurt the tree! It is a living creature
The reason why you decided instead of burying the sword is because Kolkoli saw you, and he protested to the fact that you just in fact, stabbed a living creature.