Well, I don't have Oblivion (or Shivering Isles for that matter.) But if Morrowind is any indication, I can think of three options.
1: Activate the tree like you would activate a chest.
2: Find the expert woodcutter. (is there one?)
3: Find the tool required to harvest from the tree.
Best advice I can give you is to check out the UESP wiki. I checked it out recently for my Battlespire study and I noticed a Shivering Isles section. They might have the answer you seek.