You don't need topsoil when installing sod, according to the experts with the University of Rhode Island. They suggest that you amend your current soil with organic matter, limestone and other materials. If you feel that you must order topsoil, you will need 4 to 6 inches.
One of the most critical steps to success with your new sod is proper preparation of either the existing soil or the topsoil you purchase. Submit soil samples to your county cooperative extension office to determine what is lacking in the soil and get suggestions on how to improve it for healthy sod.
Adding topsoil around trees raises the grade around them and may damage or kill the roots. Extension agents with Clemson University suggest that you construct a low wall from brick, rock, masonry or other materials around the tree.