Thanks, Joe, for the link to KIDiddles! The version of this song I grew up singing was Version #2 on that site:

I see the moon. The moon sees me
Under the shade of the old oak tree (I sing apple tree here)
Please let the light that shines on me
Shine on the one I love.

Over the mountains, over the sea
Back where my heart is longing to be
Please let the light that shines on me
Shine on the one I love.

I know that there were more verses to this version than just these two. Does anyone remember them?

Also, does anyone know who the author is of this song, or better, the original tune?

-- Áine