Is there an adjective to describe an object (edit: e.g. a dog house or a headstone; the object is not necessarily a living thing) on the ground and underneath a tree? I thought of "treeside", but its meaning is obviously a bit different.
Subarboreal is pretty close in strict definition, and could be extended to being used simply as an adjective to describe something as being beneath a tree. It is normally used to describe a species of animal or plant.
def: living or thriving under trees
See also: arboreal (leaving in/among or relating to trees)
From Latin arbor (tree) -> arboreus (of trees) and sub (beneath)
Try subarboreal. I don't see an entry for it at merriam-webster.com, but I see it used in at least one scientific paper with the meaning you have in mind.
There is also a link mentioning the word on the World English-Language Scrabble Players Association, whence I deduce (without checking) that it is in at least one officially-recognized Scrabble dictionary.