Neighbourhood disputes are hard enough to deal with at the best of times but when your property is tenanted it makes it a little more difficult as the information is often second or third hand. Tree disputes can vary from minor issues such as a tree/branch overhanging your garden and needing pruning to large branches falling on your property and causing major damage.
Generally speaking, you are entitled to cut/prune/remove any branches that are growing over your property but remember that you then have to clean up the ensuing mess. Addressing the issue with your neighbours first is advisable as they may have no idea that the tree/branch is causing so much concern or damage to your property and are often happy to deal with the matter themselves.
However, if a tree branch falls from the neighbouring property into your garden and damages your fence and your garden, then unfortunately most insurance companies view this as accidental and you will be financially responsible for the damage to your garden and the cost of the removal of the tree/branch.