I have heard it as "You have to offer them back" which makes it sound as though they can refuse them. Then you'd be stuck with them.
What about the other side of it? If a neighbour cuts his own trees severely back (or gets a contractor to do it) and some of the branches fall into your garden, shouldn't he remove them? If he can't trespass on your garden (fair enough) surely he should come round and ask for permission to collect them?
New neighbours whose garden touches mine from the side (round the corner) had one tree cut down, very near my boundary and halved the height of two others. At the same time a solid 6ft fence was put up all round. It seemed that the debris had been taken away by the contractors, until I noticed that branches that I thought were still attached turned out to be cut off and resting on top of my pergola, which runs down that side. Once they were pulled clear and cut up smaller, they filled a one-ton builder's bag. By then the contractors were long gone, and no-one was at home when I called round to ask what to do with the stuff. Because of the fence I couldn't speak informally over the wall, it would have taken a complaining visit - not a good start - so it has all been stuffed gradually into the garden rubbish wheeliebin over several weeks.