When the surrounding trees are below the level of the rooftop, there should be no problem, provided that an adequate roof-mounted UHF aerial of the correct type is used.
Where trees are above the level of the rooftop it may be necessary to install aerial so that they are above the trees. This can sometimes be achieved by mounting the aerial on a pole attached to the chimney stack or by using a remote aerial clear of trees.
The problem is more difficult for people who live in single-story houses. However, it is sometimes possible to site a receiving aerial so that it is below the main branches of the trees. If there are no other obstructions at low level, then satisfactory reception may result.
When deciding the height of an aerial, allowance should be made for growth of the tree.
If the aerial cannot be mounted above the tops of the trees, the signal will inevitably be weakened in passing through the trees. This is not necessarily a problem but it may involve the use of a high-gain aerial, possibly with a masthead amplifier. In any case it is desirable to use a good quality low-loss feeder to connect the aerial to the receiver, and it will often be advantageous to try an alternative position for the aerial - on a different chimney stack for instance.
There may be alternatives to receiving a signal via an aerial or satellite dish. Cable TV and TV received via the internet could provide the solution to reception difficulties.
The council would not prune or remove a tree simply because of its effect on TV reception.