Who is responsible for a fallen tree on a duel carriage way that hits your car?

Depending on circumstances and your policy, it would probably count as 'act of God' - no person responsible and should be fully covered on a good policy. Potentially you could argue the Highways Agency were liable if they failed to maintain the tree and allowed it to become dangerous, but that won't get far if it fell during a very heavy storm, particularly if they issued weather warnings.