The short answer is yes you can. Back to the fence line, no further. The only exemption to this is if the tree is Regulated or Significant, in which case you will need to engage an Arborist, or risk facing large fines.

You are also required to return the pruned limbs to the tree owner, but ideally we/you would have a discussion with the neighbour prior to the works being undertaken, and the foliage would be mulched and removed from site by the Arborist undertaking the work.