Hi, my moms grandparents are both from Palermo, Sicily.
My great grandmother was named Josephine Pecararo (not sure about the spelling) My great grandfather's name was Joseph Badami. They came to New York separately and met and then got married, settled in Rochester New York, where their children changed the spelling of the name to Badamy to become more Americanized as they entered the medical profession. My grandfather's name was Patrick born 1919, he had a brother Peter, both were doctors and a sister Mary who was a nurse, they were both younger than him. I think my family might be the missing branch of your tree.