Re: Appleton, NE (defunct post office)
The 1890 Nebraska State Gazetteer, Butler Co. chapter, which is online at the Butler Co. NEGenWeb site under the "Directories" link, indicates:
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"Appleton [NE] Post office established June 23, 1874; discontinued March 30, 1880. Origin of name not ascertained. There is also an Appleton, Wisconsin"
The link for Butler Co. NEGenWeb
There is also "Franklin Precinct" information from 1918 at