The song is already in the DT, as Mark indicated, with chords for the tune to which Peter Bellamy set Kipling's verses, and which is the only tune I have heard it sung to.  Bellamy's tune is given in abc and miditext formats in this thread:  Help: kipling - A Tree Song

Kipling was well aware of the folkloric background (and had almost certainly read The Golden Bough), but there is no evidence that he was "adapting" some unknown earlier text, and I fear that the suggestion that it's a modified "spell" is just the kind of wishful thinking that has bedevilled folklore studies over the years, and which makes it so hard to take many Neo-Pagans seriously.  Graves has many interesting things to say about trees, but he should not be read uncritically; The White Goddess is a problematic mix of genuine scholarship and pure fantasy.  There's really no need to posit some complicated "Celtic Tree Alphabet" connection; the tree references are all to folkloric beliefs common throughout Britain.

I have always liked Puck of Pook's Hill, and that Kipling was an adept and knowledgable versifier is clearly demonstrated by the fact that so many people imagine his work to be older than it really is, or that he must have based it on somebody else's.  Sometimes, after all, a tree is just a tree. %)