Remove any dead or damaged wood from the tree, including wood damaged by disease or heavy insect infestation. Use pruning shears for small branches and the saw for larger ones.
Examine the relative lengths of the existing leaders -- upward-growing shoots -- to determine which is the longest. Mahogany trees often have several. If two or more leaders are roughly the same length, determine which is the most central. The largest or most central leader should become the main trunk. The others compete with it for resources, creating an unstable tree.
Cut away the secondary leaders, either to their base or to a length considerably shorter than the one you have chosen to be the main trunk. The tip of the leader should be several feet higher, at least, than the tips of any other leaders.
Thin the branches of the tree to about 2 to 3 feet apart.
Prune branches as the tree grows, removing any overlapping ones, which can cause injury and allow infections or pests to become established. The University of Florida also recommends removing branches that have a diameter greater than about two-thirds of the trunk, which also creates an unstable tree, prone to topple in high winds.