Moss or lichen growing on a tree in Rhode Island or other coastal areas is a common sight. By themselves neither is likely to harm the tree.  (Neither is a parasite.)  An exception would be if the moss or lichen became so heavy on the tree that when it was wet it actually weighed down branches to the point of breakage.  But both may be indicators of problems for the tree.

I think that this is most probably an Usnea lichen which is in fact greenish algae growing on white, string-like fungus. You can clarify the identification by pulling apart a strand and looking for the white thread. This grows, in particular, on older apple trees in moist environments and is indicative of slow growth. If the tree is not an old one, the presence of the lichen may indicate that the tree is in delicate health causing growth to slow. If it does not appear to be healthy in the new growing season, you should consult a certified arborist.

If it is not lichen but moss, the presence of the moss suggest that the tree is in a moist and shady position, conditions in which moss grows best. And shade is not the best position for a fruit tree! Also, trees that are not native to cool, damp environments may not prosper in the conditions that cause moss to grow and the moss on the trunk may hide any early signs of disease. So the moss may be a diagnostic tool if this tree does not seem to be prospering once the growing season begins, indicating that the tree may not thrive in its current environment. 

With a better idea of the position of this tree in your yard, perhaps a chance to examine the green stuff more closely and maybe knowing the type of tree, you may be the most able to surmise what is going on with this tree. Some interesting detective work! I do not suggest that you attempt to remove the lichen or moss at this point.  Removal does nothing to change the conditions that cause it to grow. Wait for the growing season to determine the apparent health of the plant and you will have some helpful diagnostic tools if a call to the arborist is necessary.

Courtesy of NYBG Plant Information Service