Gilbert Pine is almost certainly descended from the immigrant James Pyne (b. 1630) and can be traced further to William Pyne, born about 1580 near Exeter, England. It is thought that Gilbert was the son and last born child of John Alexander Pine, Sr. and Hannah Hiden. I don’t know if this is proven or not. My source of much of the information I have on the family from this period is Wanda Thacker(wandathacker.50megs.com); who has extensive web pages of Pine family history.
There is also information on the Mormon Church website, FamilySearch.org, but it lists only John Pine, b. 1790, as a child of John Alexander and Hannah. The 1800 U.S. census for Bedford Co. PA lists the family of John Pine as consisting of 7 males and 3 females (6 males and 3 females ranging in age from under 10 to 25 years). The adult ages, both over 45 years, would be consistent with the age range of John A. and Hannah.
If Gilbert was born in 1810, it would mean that John Alexander would have been about 58 years old when he was born. The birth date of Hannah has not been determined, as far as I know, but was probably no later than 1755 (from the 1800 census). It has been suggested elsewhere on this forum (by Dee Triteus who has not posted for several years) that Gilbert was actually born around 1799 contrary to the age listed in the 1850 census. Ms. Triteus also posted that Gilbert was born in Bedford Co., PA also contrary to the 1850 census that lists his birthplace as Ohio. The Bedford birthplace makes more sense to me since John Alexander and Hannah do not appear to have lived in Ohio at any time and both are said to have died in Bedford. It is also possible the Gilbert is one of the 3 males age under 10 listed in the 1800 census; supporting the claim that he was born before 1800.