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After Elendil and Isildur there were eight High Kings of Arnor. After Eärendur, owing to dissension among his sons their realm was divided into three: Arthedain, Rhudaur, and Cardolan, Arthedain was in the North-west and included the land between Brandywine and Lune, and also the land north of the Great Road as far as the Weather Hills. Rhudaur was in te North-east and lay between the Ettenmoors, The Weather Hills, and the Misty Mountains, but included also the Angle between the Hoarwell and the Loudwater. Cardolan was in the South, its bounds being the Brandywine, the Greyflood, and the Great Road.
From Apendix A:
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In Arthedain the line of Isildur was maintained and endured, but the line soon perished in Cardolan and Rhudaur. There was often strife between the kingdoms, which hastened the waning of the Dúnedain. The chief matter of debate was the possession of the Weather Hills and the land westward towards Bree. Both Rhudaur and Cardolan desired to posses Amon Sûl (Weathertop), which stood on the boarders of their realms; for the Tower of Amon Sûl held the chief Palantír of the North, and the other two were both in the keeping of Arthedain.
In the case of Gondor, although there was trouble in plenty for the kingdom and it's reach was lessened, it was not divided into lesser kingdoms but remained as one kingdom with one ruler from the time of Meneldil son of Anárion through to Denethor II the last of the Ruling Stewards.