On Sat, 4 Mar 2017 09:10:34 +0000, PeterC
wrote:

This is a 'protected' sweet chestnut, but, according to Network Rail, it's
permissible to cut it if it poses a danger to people/property.

I'd check on the interpretation of posing a danger. It would become
the landowner's responsibility to take action not the neighbour but
the neighbour could apply for permission to cut back to boundary which
would normally be refused if the tree is subject of a preservation
order.

AJH