"With Love from
Dec 12 - 1911 -- age 7 1/2 mo.
To her Great Grandmother Avery."
Little Elizabeth Rountree's parents were James C. Rountree (b. 1880, Missouri) and Vera Marr (b. 1885, Missouri)1. The family lived in Webb City, Jasper County, Missouri and Elizabeth's siblings were James (b. 1908), Malcolm (b. 1918), John (b. 1921), and Howard (b. 1924)2.
Her parents James C. Rountree and Vera S. Marr were married in Joplin, Missouri on the 11th of April 19063. I found several trees on Ancestry that shows Lillian Mae Avery, who married Nicholas Marr, as being Vera Marr Rountree's mother. While I haven't confirmed this by finding a birth record, indications are good that the we have the correct Elizabeth Rountree and have discovered who "Great Grandmother Avery" is.
According to the October 1st, 2008 issue of The Joplin Globe, "Elizabeth 'Betty' Rountree Smith, a homemaker, passed away Sunday, September 21, 2008." She was the widow of Edgar Smith, and mother to the late Susan Ditty Von Till.
1 "United States Census, 1920," index and images, FamilySearch (https://familysearch.org/pal:/MM9.1.1/M8HW-PYQ : accessed 29 Aug 2012), Elizabeth Rountree in household of James C Rountree, , Jasper, Missouri; citing enumeration district (ED) , sheet 5B, family 126, NARA microfilm publication T625, FHL microfilm 1820922.↩
2 "United States Census, 1930," index and images, FamilySearch (https://familysearch.org/pal:/MM9.1.1/XHFV-RY7 : accessed 29 Aug 2012), Betty Rountree in household of James C Rountree, Webb City, Jasper, Missouri; citing enumeration district (ED) 0028, sheet 9A, family 242, NARA microfilm publication T626, roll 1205.↩
3 Missouri. Jasper County. Missouri Marriage Records, 1906. digital images, Ancestry.com (http://www.ancestry.com/search/:accessed 29 August 2012), entry for James C. Rountree and Vera S. Mann, 11 April 1906; citing Missouri Marriage Records. Jefferson City, MO, USA: Missouri State Archives. Microfilm.↩