See also

Family of Thomas BIRCH and Martha

Husband: Thomas BIRCH (1813- )
Wife: Martha (c. 1807- )
Children: Hannah BIRCH (c. 1842- )
Samuel BIRCH (c. 1844- )
Thomas BIRCH (c. 1846-1877)
John BIRCH (c. 1848- )
Jane BIRCH (1849-1882)
Marriage
There is a likely marriage in St Catherine's index - 1840 June qtr Birch, Thomas, Leeds district vol 23 page 327.

Husband: Thomas BIRCH

Name: Thomas BIRCH
Sex: Male
Father: John BIRCH (1784-1835)
Mother: Hannah FISHER (1787-1864)
Birth 4 Dec 1813 At sea on the ship "Limerick"
Information from John Birch (Birch one name study):

I too have searched the registrers for your Thomas without success, I found his birth date in the Birch family bible, held by Charles Lenny Burch of Los Angeles, California, USA. another distant cousin of mine. It seems that Thomas was born at sea. After the family had visited relatives in Ireland. The Bible reads:-
I set this page aside in this very holy Bible for the recorded events of my family. In this year of our God and King George III. 1806.
My good wife Hannah Fisher who was born in Hanging Heaton, Dewsbury. and I were joined to-gether at St Peters Church, Leeds. 15th december 1806.
We were blessed with a daughter Elizabeth who was born 16th February 1807 in our humble home in St. James Street Leeds.
Blessed with a second daughter a sister for Elizabeth on 3rd November 1809 at our home in St James Street,Leeds.
Joseph our first son born 11 June 1812 in St. James Street, Leeds. God be blessed.
Thomas our second son born 4th December 1813 at sea on the good ship Limerick. God be blessed.
Richard our third son born 14th October 1814 in St James Street, Leeds. God be blessed.
John our fourth son born 9th August 1815 in St James Street Leeds. God be blessed.

Talking of cousins if you think that Thomas is the one that you are looking for you too are a cousin of mine, Thomas was the brother of Joseph my 3rd Great-Grandfather.
Occupation 1850 (age 36-37) Cloth Dresser; Leeds
Residence 1850 (age 36-37) Leeds
Residence 1851 (age 37-38) Leeds
Census 1851 (age 37-38) Leeds
This is almost certainly the Birch family in the 1851 census in Leeds on microfilm reel HO 107/2320 folio 249:
at 6 Regent Street
Thomas Birch head aged 38 Cloth dresser born Leeds
Martha wife aged 44 born Leeds
Hannah dau aged 9 born Leeds
Samuel son aged 7 born Leeds
Jane dau aged 1 born Leeds
[Note: this entry is not in the Yorks Arch printed index but it is in their microfiche. The above is taken from the actual microfilm].
Occupation 1868 (age 54-55) Woollen Cloth Tenterer
Census 1881 (age 67-68) Hunslet
Dwelling: 6 Accommodation Rd
Census Place: Hunslet, York, England
Source: FHL Film 1342074 PRO Ref RG11 Piece 4485 Folio 80 Page 1
Marr Age Sex Birthplace
Thos. BIRCH M 68 M Leeds, York, England
Rel: Head
Occ: Tenterer (Cloth)
Martha BIRCH M 74 F Leeds, York, England
Rel: Wife

Wife: Martha

Name: Martha
Sex: Female
Father: -
Mother: -
Birth c. 1807

Child 1: Hannah BIRCH

Name: Hannah BIRCH
Sex: Female
Birth c. 1842 Leeds

Child 2: Samuel BIRCH

Name: Samuel BIRCH
Sex: Male
Birth c. 1844 Leeds
Census 1881 (age 36-37) Leeds
There are two possibilities here, but the first seems most likely:

Dwelling: 4 Foxtons Yard
Census Place: Leeds, York, England
Source: FHL Film 1342084 PRO Ref RG11 Piece 4516 Folio 9 Page 16
Marr Age Sex Birthplace
Saml. BIRCH M 36 M Leeds, York, England
Rel: Head
Occ: Engine Tenter (22.2)
Caroline BIRCH M 30 F Leeds, York, England
Rel: Wife
Occ: Cloth Weaver

Dwelling: Upper Wortley Rd
Census Place: Wortley In Bramley, York, England
Source: FHL Film 1342080 PRO Ref RG11 Piece 4503 Folio 26 Page 47
Marr Age Sex Birthplace
Samuel BIRCH M 35 M Wortley Leeds, York, England
Rel: Head
Occ: Tanner
Eliza BIRCH M 34 F Wortley Leeds, York, England
Rel: Wife
Martha E. BIRCH 8 F Wortley Leeds, York, England
Rel: Daur
Occ: Scholar
Mary Ann BIRCH 3 F Wortley Leeds, York, England
Rel: Daur
Occ: Scholar
Clara BIRCH 11 m F Wortley Leeds, York, England
Rel: Daur
Francis BIRCH W 78 M Milns Bridge Huddersfield
Rel: Grandfather
Occ: Woollen Weaver

Child 3: Thomas BIRCH

Name: Thomas BIRCH
Sex: Male
Birth c. 1846 Leeds
Death 18 Jun 1877 (age 30-31) Hunslet
In the Birch one name study this date of death is given for his father but I think it should be for this son after having searched the LDS 1881 Census for matches.

Child 4: John BIRCH

Name: John BIRCH
Sex: Male
Birth c. 1848 Leeds

Child 5: Jane BIRCH

Name: Jane BIRCH
Sex: Female
Spouse: John William SAVILLE (1846-1916)
Birth 1 Oct 1849 Green Street, Leeds1
Baptism 19 May 1850 (age 0) St Peter's (Parish Church), Leeds
Residence 1868 (age 18-19) Hunslet
Death Mar 1882 (age 32) Leeds
Died giving birth to twins.

Note on Husband: Thomas BIRCH

Lived at Grove Place, Hunslet in 1868.

Sources

1Dave Bradley, Leeds St Peter's Parish Registers. Born 1st October 1849. Baptised 19 May 1850. Jane, dau. of Thomas and.