Research the history of the palm tree you want to date. This is easier if the palm tree is on a protected site, such as a national park. A park ranger or a horticulturalist on site may be able to help you identify how old the palm tree is.
Look up the species of the tree using resources at the library to get a general sense of the lifespan of the palm tree.
Determine if the tree has gone through a radiocarbon dating process. A national park service representative might be able to tell you if this information is available. Radiocarbon dating is a scientific process that measures the tree's growth rate by analyzing it with chemicals. This is a contested method--and one that's used sometimes for dating palm trees--because it takes the tree's growth rate over a certain time span and infers this rate for the lifespan of the plant. This method, however, doesn't take into account factors that can inhibit growth at any given time, such as climate or disease.