from PIE* deru- "oak" to tree, the d became t. ( I always like that part)

Yes, tsuwm's pretty much nailed it. I'll give some other cognate words: druid (from Irish), true, Greek δρῦς (drus) 'oak', δόρυ (doru)'spear', Polish drwa 'wood'.

No to the question. Well, English tree was written down long after it had been first used. The question doesn't mean much to me. It has cognates in other IE languages, like a lot of other words in English. Our earliest records of English are from roughly the 9th century. It is thought that our word for 'tree' came from the (reconstructed) word for 'oak' in PIE (proto-Indo-European, a language we have no written records of). There are plenty of examples of this sort of semantic broadening or narrowing of meaning: e.g., deer is related to german Tier 'animal'.

Ceci n'est pas un seing.