William was "of Sanborn, New York." He was a carpenter by trade and in later life he and his wife lived in both Seattle and Sultan, Washington.
1880 Census for William Barnum in Lawrence, Kansas has Charles H. Wiseman, cousin, living with him.
From the Lockport, New York Union Sun and Journal, Monday, March 10, 1941: William I. Barnum Dies In Seattle. Ralph C. Barnum, 300 South street, received word Sunday of the death of his oldest brother, William I. Barnum, 72, of Seattle, Wash., on Saturday. Mr. Barnum was born near Sanborn and made his home here until 1905 when he went to Seattle. He was a carpenter by trade. Surviving are his wife, Mrs. Grace Barnum, and one daughter, Ruth Barnum, at home; three sisters, Mrs. Ernest E. Mahannah, North Tonawanda, Mrs. Edith Jacobus, Chicago and Mrs. Roderick Parrish, Ionia, Iowa; and three brothers, Elmer G. Barnum, Fredericksburg, Va., Henry A. Barnum of San Francisco and Ralph C. Barnum of Lockport.