Yes, the tree could most definitely be the problem. Moss thrives in moist or shady conditions (typically both). The problem is, those are exactly the conditions that lawns DON'T like, so the problem is really two-fold. The moss is invading an area, in part, because the grass is struggling to survive, leaving thin, open areas where the moss can gain a foothold. In sunny, drier areas, by contrast, the conditions are tilted in grass's favor, and moss will not generally be a problem.
If there's any way to increase sunlight and decrease moisture, do so. Can you improve drainage? Can you prune up some low hanging branches? Those steps may help. Also, raise your mowing height, and be sure you overseed the area with a shade-lawn mix, if you haven't already.If you can't do these things -- or if you have and it didn't cure the problem then you might consider turning the problem area into a bed for shade-loving perennials instead.
It's true that you can kill moss. Scotts, among others, carries a moss killer, and it works okay. But it doesn't cure the underlying issues, and The moss will eventually return, requiring repeat treatments. Rather than fight it, it's better to cure the underlying issues, or just stop trying to grow a lawn where it's not suited.
Final note: you'll often hear that moss prefers acidic conditions, and therefore you should apply lime to raise the pH. That won't hurt, but my opinion is that this isn't usually the primary problem. It's generally a combination of shade and moisture (although moss will thrive in sun too, if given constant moisture and a thin lawn).