Answer: A tree is the responsibility and property of the landowner on whose land it is growing, and that landowner may be liable for any damage caused by the tree. Tree roots, as they grow and expand — often near the surface of the ground — can certainly cause damage to paths, drives and walls.
If a tree is growing on a public footpath or highway the local council is responsible for any damage caused by the tree or its roots. This is true whether or not the council actually planted the tree — even if the tree self-seeded or was planted by someone else.
You could employ an arboricultural contractor to examine the tree and your wall, and to report on the situation. If, in the opinion of the arboriculturist, the damage has been caused by the tree roots, then with the permission of the arboriculturist you could submit that report to the council to support a claim in respect of the damage to your wall.
Alternatively you could contact your buildings insurers and ask them to investigate the damage and collect evidence to support a claim against your local authority on your behalf.
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These answers can only be a very brief commentary on the issues raised and should not be relied on as legal advice. No liability is accepted for such reliance. If you have similar issues, you should obtain advice from a solicitor.