JoeJoe wrote:
What is one allowed/not allowed to do with regards to trees in one's private
garden?

complex issue

how private? & how well known are they to the neighbours, local
busybodies; our masters?

to stay legal, issues you need to go into a

Are you in a conservation area?

Any tree preservation orders in force [hedges can have these too]?

Any restrictive covenants in your house title?

How much are you wanting to cut back? There is a 5cubic metre/year
timber cutting allowance - beyond that you must get a Forestry
Commission licence to cut [ in addition to any other permissiion
needed].

Discrete initial queries about your area and not specific to your
property are advisable. Otherwise you may alert the council and get a
preservation order or worse clamped down. If you become aware of
impending restrictions, move fast.

Ask at your local council for leaflets about conservation areas, tree
preservation orders and it may also be able to give you the Forestry
Commission leaflet about felling timber.

If you do need to seek permission to deal with the trees, application
is usually free, unlike normal planning permission

HTH