I haven't been able to view the census images for 1901 and 1911 but from what I can gather in 1901 it has born in England and in 1911 born in Liverpool? I noticed a birth index in the West Derby registration district (Liverpool, England) for a child with the surname Wood born in quarter 1 1901 who was registered without a first name. It says Wood Male on Free BMD. If you look on the England & Wales GRO website birth indexes there is no maiden surname for the mother. From past experience this usually means that the child is illegitimate and the mother's surname was likely to have been Wood. This could be the birth you are looking for.

Year birth was registered 1901
Surname at birth WOOD
Forename(s) -
Year 1901
Quarter Jan, Feb, Mar
District name WEST DERBY
Volume Number 08B
Page Number 435
www.gro.gov.uk/gro/content/certificates/