Treebeard said that Trolls were made by Morgoth in mockery of the Ents, as he had made the Orcs in mockery of Elves, however, I don't think that Tolkien meant that Ents and Trolls were related.
In letter 153 Tolkien wrote:
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I am not sure about Trolls. I think they are mere 'counterfeits', and hence ... they return to mere stone images when not in the dark. But there are other sorts of Trolls, beside these rather ridiculous, if brutal, Stone-trolls, for which other origins are suggested.
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So I think that he intended Stone Trolls such as Bill, Burt and Tom, to be entirely different from the other Trolls.