There is no restriction on which creatures can attune to magic items (but keep in mind that some items will not be usable based on the creatures body shape or ability to speak command words, etc.). Since an Awakened tree uses the stat block in the monster manual it certainly qualifies as a creature.
The relevant quotes:
Attuning to an item requires a creature to spend a short rest focused on only that item while being in physical contact with it ... the creature gains an intuitive understanding of ... magical properties
statistics appropriate for the awakened plant, such as the statistics for the awakened shrub or the awakened tree.
The only possible contention comes from the details of a short rest:
A short rest is a period ... during which a character does nothing more strenuous than eating, drinking, reading, and tending to wounds
The term characters is not clearly defined but NPCs count as characters and this tweet from Jeremy Crawford, lead designer, implies that short rests can be taken by creatures other than PCs as "companions" can attune.
The magic item attunement of your companions has no effect on the number of items you can attune to.