"branches of industry" is correct and usable in written English.
You would use "branches of industry" to refer to the different types of businesses that form an economy. For example, "Manufacturing, finance, and agriculture are all branches of industry."
The systems under consideration are differentiated in scale, ranging from countries or groups of countries, through regions or branches of industry, municipalities or industrial plants, down to single processes or equipment pieces.
But he pointed out that they were an "industrious people... now passing from mining into other branches of industry," including farming and the "building [of] the Pacific railroad".
Communication, transport, information communications, main branches of industry will grow.
The main branches of industry are engineering, metalworking, and paper-pulp making.
Machinery condition monitoring is rapidly finding applications in all branches of industry.
Under communism the principal branches of industry, services, and trade were directly owned by the state.
Compared with other branches of industries, information technology in civil engineering does not reach the desired extensiveness in its application.
The major branch of industry has been heavy engineering, dominated by a huge diesel-locomotive works.
Each branch of industry and administration had its own training schools, from which it selected qualified students with satisfactory records.
Frank Zappa, the musician who died in 1993, used to say that "Politics is the entertainment branch of industry".
This branch of industry is typically confronted with high exposure to PAH and with PAH-induced occupational (cancer) diseases.
