George Trautvetter, his wife and children immigrated to the United States in July of 1853. The family apparently made a relatively direct route to Hancock County, Illinois, where George appears to have paid cash for a farm in the northern part of that county's Rocky Run Township.
Trautvetter made a declaration of intention to become a United States citizen in January of 1855 (part of which is shown in this blog post). If he completed the naturalization process, it was not done in Hancock or Adams Counties in Illinois and, given that he apparently lived in Hancock County until his return to Germany in the 1869/1870 time frame, it would have been unusual for him to have traveled elsewhere to naturalize.
George's sons, Michael, George, and Theodore were all born in Germany before their family's immigration to the United States. If their father had naturalized and if they had been under the age of majority at that point in time, they would have become citizens via their father's naturalization.
George returned to Germany (by himself) in 1869/1870 where he died. His wife and children remained in the United States.
There are no naturalization records for the Trautvetter boys in the United States--at least none in Hancock County where they lived from 1853 until their deaths in the early 20th century. Census records for the Trautvetter boys indicate that they were United States citizens. There's no record of them ever having naturalized in their own right and their father apparently never naturalized.
It's possible that they thought their father was naturalized by the 1855 declaration of intention and that, since they were underage at that time, they were naturalized as well.
Possible.
Some immigrants were confused by the naturalization process--especially in cases where the citizenship status of the father was not clear. It's also possible that George (the father) intentionally never completed his citizenship status--particularly if in the back of his mind he thought that some day he may return permanently to Germany.
Possibly.