If you are talking about one specific tree, known to both you and the listener, you could say this. For example, if you are talking about your garden, and there is only one tree in this garden, then you could say 'I can/can't climb the tree.' In other situations, you might say 'I can't climb a tree' (meaning any tree), or 'I can't climb trees' (meaning a number of trees, or trees in general). It all depends on the context. And the more context you can provide, the better our answers will be!

Hi! Yes, both are correct.

Yes. "I can climb the tree" and "I can't climb the tree" make perfect sense (in the right context of course!).