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I am afraid that your neighbour has acted correctly.
He is legally entitled to remove branches overhanging his property but should return "your property" (the branches) to you.
It is you I am afraid who have to dispose of them - although it would of course have been more neighbourly for him to have done it
Sorry to have to give you bad news please ask if you need further details
This is based on the argument that you can report your neighbour for fly tipping since he did not ask you first.
This would still leave you responsible for disposing of the branches (they are yours) - but he might be fined for the offence