Since the only sure way to age a tree is by cutting a tree down and counting growth rings or by taking an invasive trunk bore and counting rings, this short-cut method will give you an approximation.
Step 1: Determine the tree species. Some species may be grouped such as the White Oak species may be used for other members of the White Oak Group such as Bur Oak and Swamp White Oak.
Step 2: Determine the diameter, in inches, of the tree at 4.5 feet about the stump level. This can be most easily done by measuring the circumference with a tape measure at that height and then dividing the circumference by 3.1416 to arrive at the diameter in inches.
Step 3: Locate the growth factor for the tree species in the growth factor table and multiply that by the diameter in inches to arrive at the approximate age of the tree.
Forestry people have determined that there are differences in growth between forest trees and landscape specimens. Landscape trees tend to grow faster due to better care and less crowding. Therefore landscape trees may not be quite as old as the calculation indicates and forest trees will grow slower and thus may be older than the calculation indicates.
Also, since young trees grow faster, you should slightly reduce the growth factor and for very old trees which have grown slowly in old age, you should slightly increase the factor.
And remember - this is an approximate calculation, not the definitive answer.
If you have a Red Oak and find that the circumference at 4.5 feet height is 69 inches, then dividing by 3.1416 gives a diameter of 22 inches. In the table, the growth factor for Red Oak is 4.0. Multiply 4.0 times the 22 inches and the result is an approximate age of 88 years.
Thanks to the Morton Arboretum for providing the Growth Factor table.