The main focus will be on Chesterfield Avenue where 45 trees, including 32 Horse Chestnut, eight Beech, four Lime and one ash will be cut down.
The felling is part of a tree survey programme involving the 20,000 trees in the Park.
The Office of Public Works said diseased and dangerous trees have to be felled, if they could endanger people travelling on the roads.
Environment Minister John Gormley said in excess of 5,000 new trees have been planted over the past two decades and it is planned to plant a further 500 this winter.
