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Tom Bombadil says this to the hobbits, in reply to the question "who are you?":

(...) But you are young and I am old. Eldest, that’s what I am. Mark my words, my friends: Tom was here before the river and the trees; Tom remembers the first raindrop and the first acorn. He made paths before the Big People, and saw the little People arriving. He was here before the Kings and the graves and the Barrow-wights. When the Elves passed westward, Tom was here already, before the seas were bent. He knew the dark under the stars when it was fearless – before the Dark Lord came from Outside.

So basically, he was here before everything else. After Helm's Deep, when a party rides to Isengard, Gandalf says this to Théoden:

Well, Théoden, will you ride with me to find Treebeard? We must go round about, but it is not far. When you see Treebeard, you will learn much. For Treebeard is Fangorn, and the eldest and chief of the Ents, and when you speak with him you will hear the speech of the oldest of all living things.

Maybe this is just an inconsistency not intended by Tolkien, I don't know. A few years ago, I read somewhere that Tom Bombadil is not "a living creature" in a traditional sense, but there wasn't a strong argument supporting that. What is your take?

Yavanna had created the trees and Melkor was destroying them. So to help Yavanna, Iluvatar sent spirits with which to ensoul certain trees to serve as protectors of the others: these creatures were the Ents. Treebeard was the oldest of the Ents, and so the firstborn of all sentient or perhaps animate things on Arda, or as Gandalf loosely says, 'living creatures'. Trees and other plants did come before Ents. This is all in the Simarillion.

That takes care of Treebeard. What of Tom Bombadil?

In the entire canon of LOTR/Silmarillon (and to my knowledge all Tolkien's writings but then I haven't read them all now, have I?) there is no in-universe explanation* of who or what Bombadil is. A Maia? A Vala? Eru himself? Some other kind of being? All sorts of theories have been proposed. But in the stories? As Goldberry says, "He is".

So if Bombadil told the truth when he said he was older than the seeds of the trees, then he was older than Treebeard. But he may have lied. Or was mistaken. Or Gandalf lied. Or was mistaken. Or Bombadil may not have been of Arda but of the Heavens. The stories don't say.

*Out-of-universe, Tolkien had already invented the character of Tom Bombadil prior to LOTR and included him as "an 'allegory', or an exemplar, a particular embodying of pure (real) natural science..."

From Wikpedia about Tom Bombadil:

Treebeard calls himself the eldest living being of Middle-earth and says that he was there before anyone else. However, Tolkien remarked in another context: "Treebeard is a character in my story, not me; and though he has a great memory and some earthy wisdom, he is not one of the Wise, and there is quite a lot he does not know or understand."

Treebeard as an Ent was created as part of the Music of the Ainur, and the awakening of the first Ents is precisely datable to the same time as the Elves awoke, as is told in the Silmarillion chapter Of Aule and Yavanna:

'Yet it was in the Song', said Yavanna. 'For while thou wert in the heavens and with Ulmo built the clouds and poured out the rains, I lifted up the branches of great trees to receive them, and some sang to Ilúvatar amid the wind and the rain.'

And:

Behold! When the Children awake, then the thought of Yavanna will awake also, and it will summon spirits from afar, and they will go among the kelvar and the olvar, and some will dwell therein, and be held in reverence, and their just anger shall be feared.

This means that Treebeard's origin is at the very earliest Year of the Trees 1050, just immediately after Varda's second star-making.

If we take Tom's claims at face-value:

Tom was here before the river and the trees; Tom remembers the first raindrop and the first acorn.

This dates him to at the latest the time of the Valar's dwelling on Almaren and the time of the Lamps, because during that time (Silmarillion, Of the Beginning of Days):

Then the seeds that Yavanna had sown began swiftly to sprout and to burgeon, and there arose a multitude of growing things great and small, mosses and grasses and great ferns, and trees whose tops were crowned with cloud as they were living mountains, but whose feet were wrapped in a green twilight.

Since this was before the Years of the Trees, Tom Bombadil is therefore oldest (although whether he - whatever he is - could be called "living" is another matter).

Page 457 of the Lord of the Rings Treebeard speaks of the old Forest saying:

"there are hollow dales in this land where the darkness has never been lifted, and the trees are older than I am"

Hope this helps.

I believe Tom Bombadil is the older of the two. I remember it saying that the elves taught the trees to speak. Now I know the trees are different from Ents, but I assume that the elves would still have some association with the making of Ents. Now if this is true, then Tom Bombadil must be the older, for he was there before the coming of elves, and therefore before the trees and ents. That is if I'm correct in saying that the Ents were made around the time the elves taught the trees to speak. This is a total guess on my part.