If there is a TPO (Tree Preservation Order), then permission usually has to be sought from the local planning authority to maintain or remove a tree. Lots of red tape, in many cases for good reason, but where there is an obvious danger, or the tree is dead or damaged already (Much more likely to fall or be blown down), then a good case can usually be made. https://www.gov.uk/government/uploads/s ... 127793.pdf
If the tree is on someone else's land but not protected by a TPO, then the rules are more straightforward and any part of the tree that overhangs may be trimmed back. (Of course this can unbalance a tree and cause it to be more susceptible to collapse), or action taken against the owner.
The type of subsoil is likely to be relevant to the likelihood of a tree falling too, especially if it is say on a steep embankment (although sometimes roots can stabilise and remove moisture in such circumstances). Ensuring that unstable soils are well drained is vital to prevent landslips, and this applies to soils and rock generally. (Think Hatfield Colliery a few years back and how that affected the Doncaster - Barnetby line in 2012?, or the Aberfan disaster back in 1966, - not railway related, but the fact that Hatfield happened, suggests some lessons were not properly learnt or carried forward, despite the mass rush to level coal tips in the aftermath).
Trees cut back to base, usually leave the roots in place and these commonly regrow unless treated properly at the time, creating the same problem in years to come.
Further reading for those interested could include:-
Foundations of Engineering Geology by Waltham and published by Spon ISBN 978 0415 46960-9
Tree Roots in the Built Environment by Roberts, Jackson and Smith published by the TSO. ISBN 011-753620-2
Leaving matters for maintenance in the future rather than being dealt with properly and comprehensively by design and action at the time is a often false economy, as the excuse of 'no money' will later be utilised to avoid the necessary costs.