It is nearly always the responsibility of the land owner to ensure the safety of trees. In some occasions a rental agreement may put the responsibility on the tenant to manage the trees.
The legislation that effects the management of trees comes under Common Law:
A tree owner or manager has a (Common Law Duty of Care) to:
Take reasonable care to avoid acts or omissions which they can reasonably foresee would be likely to injure their neighbour.
Occupiers Liability Acts
The owner of the land has a duty of Care under the Occupiers Liability Acts to ensure that they take reasonable steps to ensure that visitors or trespassers on their land are safe.