Swedish thinks of the mountain as a surface you're on when you climb it, but the tree as something that encloses you while you climb it, so you're in it. There's something similar in English: if you describe your position while doing those actions, you'd probably say you're on the mountain but in the tree.
You can't use klättra and berg together without a preposition.
On the other hand the verb besteg is used without a preposition: Mannen besteg berget also means 'The man climbed the mountain', and you can't add a preposition there.
You can also use the verb klättra without an object, and then of course no preposition is used either.
Is there a distinction between climbing a mountain to the summit, and just climbing on a mountain but not actually getting to the top? And if so how does one distinguish in Swedish. In English we would only say "climbed the mountain" if you reached the top, whereas "climbed on the mountain" would suggest you did not reach the summit.