Branches from our neighbour's tree overhang into our yard causing considerable mess.

Answer

You need the owner's permission to take the fruit. Since the fruit rots away, he will readily grant permission. Avocados are expensive fruit. If he gives permission, you may either use it or give it away. If he intransigently refuses permission, then collect the fruit and send it to him. If he refuses permission to cut the branches, you may go ahead and cut them, but it will create considerable ill-feeling with the neighbour.