Greene is a town in Chenango County, New York, United States. The population was 5,604 at the 2010 census. The town is named after General Nathanael Greene. It is located in the southwestern corner of the county and contains a village, also named Greene. The town and village are northeast of Binghamton. Greene was founded in 1792, but com…Greene is a town in Chenango County, New York, United States. The population was 5,604 at the 2010 census. The town is named after General Nathanael Greene. It is located in the southwestern corner of the county and contains a village, also named Greene. The town and village are northeast of Binghamton. Greene was founded in 1792, but commenced in 1798.
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