And so is the book chapter I referenced -- it takes place in underground tunnels underneath mountains. But I suppose the problem lies with the English; "underground" when used as a noun more commonly denotes "resistance movement". (See definitions
here or
here) Normally you use it as an adjective ("underground river") or an adverb ("river flowing underground").
That's why both "The Giant Tree's Underground" and "The Underground of the Giant Tree" sound so weird to me -- you don't commonly see "underground" used in such a manner.
But on the flipside, "Beneath the Giant Tree" doesn't explicitly show that it's below the ground (it could be at the
base of the tree). It still sounds more natural to me, though. I say, let the audience work out the context, and put a note if need be.