It could be the lack of string tree, but I think you should check the nut slot too.
If you def need a tree, I think most are ok, just choose what looks best. If you are worried bout the string sticking in the tree you can use some lubricant like Big Bends Nut Sauce on a non-roller type, or get a graphtech one if you don't mind the non-metallic look.
Alternatively (and a lot more expensive) you could get staggered tuners so you will get more of an angle over the nut.