The term "sprite" is defined variously by different authors. Some times the term sprite refers purely to water sprites, which are generally small, wingless, hominid water faeries in modern literature. However, in medieval times, water sprites were thought to be human sized, as with the famous Melusine myth, for Melusine's mother was sometimes called a water sprite. After that, a sprite can be defined as any small faerie, generally winged, except with the exception of water sprites, who are generally wingless. Take, for example, the Spiderwick Chronicles. In that series, sprites are any faeries who look like any combination of people, insects, and flowers.
I'm not particularly familiar with the term "tree sprite" so I couldn't tell you whether or not this is a sprite that is attached to a tree in a manner similar to that of a dryad, or if it is simply a sprite residing in a tree.
You must also keep in mind that these terms could be creatures not similar to those of folklore of the author or the show's own invention. In a made-up world, things are however the author desires.