Yes, different trees. It does assume a normal distribution, but I think that's at least in the ballpark.
I'm not sure, but if going from k=1 to k=2+ produces a significantly different result, that seems strange. I'd probably spend some time trying to suss out bugs. Like what if you do k=2 and only consider the 2nd point, so there's still only 1 point in your kernel density? Or what if you do k=1 but then feed two of that point into your kernel density?