Tolkien is pretty much low fantasy except for the Silmarillion
Tolkien was a the grandfather of high fantasy.
It is often wrongly assumed that the terms high and low fantasy refer to the prevalence of magic in the setting. What they actually mean is primary v. Secondary world settings.
High fantasy takes place in self contained settings that have no relation to the real world. This includes The Lord of the Rings, A Song of Ice and Fire / Game of Thrones, and Mistborn. Even though LotR is presented as a fictional history of our world, because it has no geographic or historical similarities, it is high fantasy.
Low fantasy is when characters from our world cross a threshold into a magical world. This includes The Lion the Witch and the Wardrobe, Alice in Wonderland, Neverwhere, and Harry Potter.
