It depends on whether you live in a neighborhood with a development approval or not. Most formal developments and subdivisions in Town have a development approval which includes an approved Site Plan and Landscape Plan of record on file with the Town. Most of these neighborhoods have a Homeowners Association (HOA), which should be able to answer your question. However, when in doubt, please contact Natural Resources.
ANSWER (a): If you live in a neighborhood with a development approval and a HOA, and the trees or vegetation are located on your lot of residence, you do not need Town approval, only HOA approval. However, there is an exception to this: if the tree in question is a street tree (usually located within the green space between the sidewalk and edge of road) or is a tree or vegetation on a common area, then Town approval is required. In these cases, the HOA should apply to the Town for a Vegetation Removal Permit.
ANSWER (b): If you do not live in a neighborhood with a development approval or a HOA - these are older neighborhoods (see list below), often referred to as "charter" neighborhoods - you do not need Town approval to remove trees or vegetation if they are located on a developed lot. These older neighborhoods are exempt, because at the time of development, there were no requirements for a minimum amount of preserved native vegetation or landscaping. If the lot in question is undeveloped, or is not zoned Residential, please contact Natural Resources.
The older neighborhoods are: Pine Gardens North, Pine Gardens South, East View Manor, Jupiter River Estates, Penn Park, Brentwood North, Brentwood South, Riverside Drive Park, Olympus, North Palm Beach Heights, and most streets north/south of Center Street.