The unnamed man was found on the Southampton City Council-run Southampton Common last Wednesday afternoon (6 August).
Firefighters lifted the branch using specialist lifting equipment in an attempt to rescue the man, but he later died in hospital.
The council has confirmed it was responsible for the tree, and has begun an inquiry. The Health & Safety Executive is also investigating the case.
A council representative added: "This was a mature oak in woodland away from marked tracks. We manage 55,000 trees in the city and as our resources are limited, this would not have been inspected as regularly as others."