Oh dear, sorry Tinks.
Like Staggers said a neighbour can only legally lop off any branches off your tree if they overhang your property, and they do have to offer them back to you, not throw them in your garden!
When I lived in London I had a sycamore tree in my garden. My neighbour had a fear of insects and thought my tree attracted them, even though none of it over hung their garden.
She called a tree surgeon and told me I could have it removed for a hundred pounds. I told her I liked the tree because it afforded my garden some privacy from the houses behind.
On returning from holiday I discovered the insect hating nutter had paid the tree surgeon herself and the tree had gone completely.
I won't write what I said to her, because the swearing censor would go into overload.
The only positive thing was I didn't have to spend ages weeding out the seedlings from the sycamore, sent spiralling out on it's little spinning jennys.