Is it obligatory to seek forgiveness from someone whose valueless (in common terms) possession, such as a tree leaf, was damaged, and is not seeking forgiveness in such matters the correct thing to do?

There is no objection to cutting tree leaves without permission, as long as it does not harm the tree, because custom has shown leniency in this regard. Requesting permission might lead to misunderstanding or accusing the person asking of being overly strict.