Useful posts above, thank you.
Just to clarify, if you are removing dead branches from live trees, or dead branches from dead trees but "don't go there" , you are not obliged / required to 'serve notice.'
However, as also advised, and particularly if it's in a deemed sensitive area, a quick call, and ideally email, to the Planning Dept / Tree Officer would be helpful / courteous.
Cheers..
Paul
PS Also remember, particularly on Oak, there's lots of little beasties and wrigglies that need that aerial deadwood for habitat etc. so go easy