DearCustomer thanks for your prompt answer.
I accept that, in the absence of any agreement to the contrary, he is obliged to return the cut branches to me. However, surely that is just in case I actually wanted them, for timber etc.
Here, I have told him that I do not want them. In that case, surely I do not have to accept them? He did not have to let them grow so big, on his side of the boundary. By doing so, surely they are his liability?
I have read elsewhere that if I have told him not to put them on my garden, then he is no longer entitled to, and he may be guilty of fly tipping.