It took some looking to get started.It helps if you can give more info, such as when and where they lived, names of children, etc.
I found a William Burklin Carter on the WWI draft registration in NC.
Knowing that NC has many vital records indexed on Ancestry.com I started there.
I found Lizzie Strickland married Benklin Carter, Dec 22, 1907 in Wayne County, NC.
Using Williams age and location from the 1917 draft registration, I located their family in South Clinton,Sampson County, NC on the 1920 census.That gave me Lizzie's age, 31, and place of birth NC.
Then I was able to locate a copy of the death certificate for Sophie Elizabeth Carter.
She died August 14, 1962 in Clinton, Wayne County. She was born Oct 6, 1888, daughter of David Strickland and Catherine Wiggins.She was buried in Epworth Cemetery Aug 17, 1962.The death certificate reported she was a life long resident of Wayne County.
On the 1900 US census, Her family is living in Mount Olive, Wayne County, NC
Her father David, was born Dec, 1856.
Her mother Millie C, was born Jul 1867.They married about 1885. Each was born in NC as were their parents.
Siblings were Thomas, Apr 1887, George, March 1890, Lewis, Feb 1892, David, Jun 1895,
Then I found the Death Certificate for Katherine Wiggins Strickland.She was born July 30, 1867 in NC, daughter of Thomas Wiggins, b. NC and Sophie Carroll, b. NC.Katherine died May 6, 1916 in Goldsboro, Wayne County, NC.She was married at the time of her death. She was buried in Mt Olive, NC.
On earlier census, there is a David W Strickland, b. about 1856, living in Sampson and Wayne Counties.His parents are Edward, b. abt 1814, and Elizabeth, b. abt 1814.
On the 1870 census, the Wiggins family was living inPiney Grove, Sampson County, NC.Thomas was born about 1822, and Sophia Jane was born about 1828.Matilda Catherine was age 2, born about 1868.The Strickland family had moved to Piney Grove by the 1880 census.
I had to look further to find David's death certificate.I ended up finding him in Sampson County.He was found dead of apparent heart attack in Mingo township, Sampson County, NC on Jan 1937.He was the son of Edward Strickland, and unknown mother.No birth date was given, but age was reported as 80.Burial was in Goldsboro.
I'm not related.