What was the tension between Native Americans and White Settlers during the years 1800-1850?In other words what was the problem and the background of the conflicts at that era?

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The source of tensions between Native Americans and white settlers, of course, was the fact that the United States was aggressively spreading into Indian lands west of the Appalachian Mountains. In the late eighteenth century, the US government had conducted an expensive and bloody campaign against Indians in the Northwest Territory that concluded with an American victory at the Battle of Fallen Timbers in 1794. Many Native Americans  in the region were not party to the Treaty of Greenville that followed the battle, however, and resentment at continued white expansion was greatly resented. In the early nineteenth century, a movement began under Shawnee leader Tecumseh and his brother Tenskatawa, or the Prophet, to united various Indian peoples together (with British support) to roll back American expansion. The Prophet was defeated at the Battle of Tippecanoe in 1811, however, and Tecumseh, an officer in the British Army in the War of 1812, died at the Battle of the Thames in that conflict. The dream of pan-Indian resistance in the Northwest died with him, and as whites moved into the modern upper Midwest, Indians either fled to Canada or moved onto reservations.

In the South, powerful Indian tribes continued to exist even after the Revolution, but many, particularly the Cherokee, embraced the US government's policy of "civilization" as the best means of securing their survival. The major tribes, who became known as the "five civilized tribes" were the Cherokee, Chickasaw, Choctaw, Creek, and Seminole. This decision came after military defeat in some cases, like the Creek Red Stick War and the Seminole War, but many people in these tribes, especially elites, adopted white dress, agriculture (including slavery) and government. The Cherokee, in particular, set up a constitution not unlike that of the United States. But the cotton boom caused continued demand for Indian lands by southern whites, who desired the fertile lands inhabited by Indians, who they increasingly characterized as racially inferior. When Andrew Jackson, who had participated in both the Red Stick and the Seminole Wars, was elected President in 1828, he adopted a policy of Indian removal. One by one, beginning with the Choctaw and ending with the Cherokee, the Southeastern Indians were removed to Indian Territory in modern Oklahoma. 

The cause of tension between the United States and Indian peoples, then, was the implacable desire among Americans for western lands. This would continue to the Civil War, when conflicts between American troops and Cheyenne, Sioux, and Apache fighters broke out in the Great Plains and the Southwest. Similarly, whites encroached on Nez Perce lands, to name just one group, in the Pacific Northwest.