From a legal site
It is an offence to top, lop, uproot, wilfully damage or destroy any tree in a Conservation Area without having first given notice to the local authority, or any tree already subject to a TPO. There are limited exceptions, such as where a tree is dead or dying, dangerous or a threat to life or property. The Magistrates Court has power to impose fines of up to £20,000 per tree, but if the proceedings are brought in the Crown Court, there is no limit to the fine which can be imposed. Felled trees must be replaced.
A property owner is responsible for any tree on their land, in the same way that they are for anything else on their land. A prudent property owner will take reasonable action to ensure that no tree on their land will cause harm or damage to any neighbouring land or buildings. A wise property owner will also avoid things which may cause friction with neighbours. Apart from the many other considerations, legal action, whatever the form, can be extremely expensive.
If your tree has branches or roots which encroach on your neighbours property, your neighbour is quite within their rights to cut back the encroachments, from their side, without giving you any notice or warning, although they are obliged to return the cut offs to you. This is a right that exists under ‘common law’ unless precluded by something else, such as the provisions mentioned above in respect of Conservation Areas or by TPO. However your neighbour in exercising this right is also responsible for the action they take and any damage caused, for example if their action were to affect the tree’s stability.
Furthermore, if the roots from your tree encroach on your neighbours land and cause damage to their property, you can be potentially liable for this. Common problems are damage to drains, or subsidence in buildings. You can be liable for the damage caused even if you were not aware of it, if this was reasonably foreseeable, and you could have taken reasonable steps to prevent this. The responsibility is on the owner if they know or ought reasonably to know of the damage, and have a reasonable opportunity to deal with the issue. Actions could be brought for the costs of repairing damage, or in urgent cases your neighbour could seek an injunction.
If you have a number of tall trees forming a hedge, and these are evergreen, over 2 metres high and detract from a neighbours reasonable enjoyment of their house or garden, your neighbour can complain to the Council under the Anti Social Behaviour Act 2003. The Council can make an Order to reduce the height or otherwise impose requirements for its management. If any tree is posing an imminent danger, the Council also has power to make an immediate Order against the property owner requiring work to be done to a tree
Property owners should regularly survey trees with the benefit of professional advice, to check whether they are causing damage, or likely to cause damage to surrounding property. Tree owners should be prepared to manage trees properly to reduce the risk of damage or harm, and any possibility of liability. The costs of proper management are monies well spent compared with the vast potential for liability if ever court action were taken.