Tree roots are opportunistic in their growth and do not respect legal boundaries. They will grow under walls and, in some circumstances, under paths and roads. They don't necessarily grow evenly around a tree and may grow suprisingly deep. Not all tree roots are the same. As a rough rule of thumb many broadleaved tree roots will be at least as extensive as the crown of the tree, but how extensive varies with the tree species.
Tree roots can affect property and infrastructure in two main ways:
Directly: for example a root grows into a drain, or presses against a wall or
Indirectly: the tree root dries the shrinkable soil (normally clay but sometimes peat) beneath a wall or foundations. Whilst clay soil is common in parts of England, there are areas where parent material (i.e. the rock underneath the soil) is less likely to produce clay soil. It is important to first undertake some basic tests before leaping to conclusions about whether a tree is causing problems. The British Geologic Survey provides basic geological information but it may take a specialist knowledge to interpret data collected on soil conditions.
If you have received a letter suggesting the roots of your tree may be causing problems to your neighbour, you may need professional advice unless you are willing to undertake the tree work suggested. Your ignorance of the problem to date is likely to be a reasonable defence. However, once informed of the problem you need to act in a reasonable and timely fashion to avoid legal action against you. This may involve seeking professional advice, contacting your liability insurers (normally part of your contents insurance) or just removing the tree(s). However, in some, but not all, circumstances, removing a tree may lead to further damage as a result of heave. You are advised to seek professional advice in these circumstances as buildings may be affected in different ways by the removal of tree(s).