Is there any tree preservation order on the tree?
You can cut off any overhanging branches if you wish but you mustn’t keep them because they don’t belong to you.
If the branches are causing nuisance and you want to get someone else to cut them because you can’t do it yourself, provided you give the neighbour warning that you are going to do that, you can seek the costs from your neighbour.
You ask about responsibility of the branches.
It depends on whether any damage, even in a storm, was reasonably foreseeable. If the storm was of such magnitude that widespread damage could not have been foreseen, then the neighbour can escape liability. If however a reasonable inspection of the tree would show that it was diseased or decayed, then the neighbour would be responsible if they had not had that inspection carried out.
If they did have the inspection carried out but it was defective, then liability would attach to the neighbour and they would pass it on to the tree inspector.
Can I clarify anything for you?
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It doesn’t matter whether it’s a neighbour or an estate except that they may have more money then you have to deal with any litigation.
The branch which has blown off looks like a healthy branch to me which has been broken by some kind of storm and hence, if someone was injured by a branch like that, that kind of damage would not be reasonably foreseeable and hence not actionable.
The problem for the estate is that if you put them on notice that this tree is potentially dangerous, they are then under a duty to inspect it.
I suggest you ask them for the date of the inspection and a copy of the report. They don’t actually have to give it you until you issue a court application for pre-action disclosure in anticipation of any litigation, but if there is a problem with this in the future, it would be good to have the correspondence and their reply to support your case.
I would also suggest that you ask the qualifications of the person who inspected the tree and when the inspection was undertaken.
It was my pleasure to assist you with this. Best wishes.