Eldest?

Who is the eldest? Bombadil or Treebeard? This question just occured to me a few hours ago as I was rereading The Two Towers. This is from chapter 5 The White Rider.

Treebeard is Fangorn, the guardian of the forest; he is the oldest of the Ents, the oldest living thing that still walks beneath the Sun upon this Middle-earth.

Bombadil himself quite clearly states that he is the eldest, and this is in seeming contradiction. I am afraid I cannot offer much more than this question, my own feeble reasoning is that Tolkien is implying that Tom isnt exactly a living thing. He certainly is alive but I think not in the sense of the Children of Eru or the free creatures (which may shed a bit of light as to his nature). He is his own. That is all I have to go on, terribly sorry it is not much.