The age of the living trees is found, with the help of an instrument called an 'increment borer' with relatively minor injury to the tree.
It is a type of drill that is inserted into the trunk of the tree at its widest girth and to extract a section of wood tissue from a living tree.
Thus we can obtain a linear core of wood which can be considered as the number of annual growth rings.
Image Source: http://en.wikipedia.org/wiki/Increment_borer