Born in Fort Worth, Texas, Buxton graduated from Central High School and then joined the U.S. Naval Air Force. After his military service, he earned a B.L.A. from Texas A&M University in 1950. He became a registered landscape architect in Texas and, in 1956, established the landscape architecture firm Fred Buxton & Associates in Houston. Specializing in parks, plazas, campuses, and water conservation systems, his firm employed many architects and landscape architects who became well-known in their own right, including Lanson B. Jones and Charles Tapley. In 1971, Buxton was commissioned by the University of Houston to design the Student Life Plaza with architect John Zemanek, and again in 1972 to design the Cullen Family Plaza with landscape architects Cornell, Bridgers and Troller. In 1976, Fred Buxton & Associates won a design competition held by the Junior League of Houston for Market Square Park. Occupying a city block in downtown Houston, the park was dedicated as part of the U.S. Bicentennial. Buxton developed master plans for sites throughout Houston, including the 324-acre Texas A&M Research Park and the Texas Medical Center. He became the landscape architecture consultant for the land-planning firm Knudson & Associates in 2002. Buxton was a member of both the American Society of Landscape Architects and the American Planning Association. In 1983, Market Square Park was listed as a contributing feature of the Main Street/Market Square Historic District in the National Register of Historic Places. Buxton passed away in Houston at the age of 78.