Yes, see Ancestry, "Randy Lineage" by "frankie49" + others' trees link them just as you say. MY THEORY on next earlier generation supported by tax & census is: Thomas Basket as father not proven, but this is likely the Family:
Thomas Basket b.ca1785 SC m. ___unk____ b.ca1785 GA. Children:
--1. William Basket b.1820 TN m. Mary P Turpin b.1821 AR dau of Elisha Turpin & Mary Bayless.
--2. Thomas Basket b.1828 MO m. Harriett Yount b.1829 MO.
--3. Rebecca Basket. b. 1832 MO.

See the following tax & census records that suggest this relationship:

1828 Gasconade Co MO Tax List [Gasconade County, Missouri):
Thomas Basket
John Turpin
Josiah Turpin came 1813.


1830 US Census Gasconade Co MO. Thomas Basket:Males:1:0-5, 1:5-9,1:30-39;Females:1:15-20,1:30-40; total: 5.

1840 US Census Ripley Co MO (Ancestry pg 7/32)
-Thomas Basket: Males: 1: 10-15, 1: 50-60; Females: 1: 5-10, 1: 15-20, 1: 40-50; Total: 5.
-Benjamin Turpin (Tirpin): Males: 1: -5, 1: 20-29; Females: 2: 15-19, 1: 20-29, 1: 50-59; Tota:l 6.
-Martin Turpin: Males: 1: 20-30; Females: 1:20-30; Total: 2.