Dacorum Borough Council states the conservation area rule plainly: "All trees in conservation areas are protected if they are more than 75 millimetres in diameter, measured at 1.5 metres above ground level", and "It's an offence to work on a tree in a conservation area without giving us six weeks' written notice (known as a Section 211 notice). This period allows us to decide if the tree(s) needs protection through a Tree Preservation Order". It lists the exemptions as dead trees, dead branches and twigs, and dangerous trees, and adds "you should check with us first, giving at least five days' notice".
On orders the council says "A TPO makes it illegal to fell, lop, prune, or damage a tree without our consent", that it can make an order without notice or consultation, that there are at least 28 days in which to object, that replacement planting will be required if removal is approved, and that "All works to TPO trees should follow the British Standard 3998". Dacorum has 25 conservation areas, and the council names Berkhamsted among the large central areas.
Nationally, gov.uk describes the requirement for "people to notify the local planning authority, using a section 211 notice, 6 weeks before carrying out certain work on such trees, unless an exception applies", and says the work may go ahead sooner where the authority gives consent. The same guidance states that consent "is needed for carrying out work on diseased and/or dying trees unless some other exemption applies".
We would explain how you check the conservation area boundary, any tree preservation order and the proposed work before confirming whether a section 211 notice or consent is needed.