Question

Which branch of science was newly developed after the discoveries about DNA?

Answer 1

I would say molecular biology most probably 

Answer 2

You can't say it developed just one branch of science.
To name a few: Genomics, Genetics, Chemistry and The science of studying DNA itself. We can't forget about the uses in technology like DNA Nanotechnology.
But technically, the main branch of science developed after the discoveries about DNA was Molecular Biology.