
der Zweig
branch / twig
A smaller, woody side shoot that branches off from a branch of a tree or shrub.
Ein Vogel saß auf einem dünnen Zweig und sang.
A bird sat on a thin branch and sang.
branch
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branch / twig
A smaller, woody side shoot that branches off from a branch of a tree or shrub.
Ein Vogel saß auf einem dünnen Zweig und sang.
A bird sat on a thin branch and sang.

branch / field
A subfield or division of a larger structure, such as a science, an art, a family, or an organization.
Die Robotik ist ein schnell wachsender Zweig der Ingenieurwissenschaften.
Robotics is a rapidly growing branch of engineering.

sector / branch
A specific sector of the economy; an industry or economic sector.
Der Tourismus ist der wichtigste wirtschaftliche Zweig auf der Insel.
Tourism is the most important economic sector on the island.
You can recognize the meaning of 'Zweig' (branch) from the context. If you are talking about trees or plants, the literal meaning (m01) is intended. In connection with science, art, or family, it is a subfield (m02). In the context of business or industry, it is a sector (m03).
Im Frühling bekommen die Zweige neue Blätter. (Natur)
In spring, the branches get new leaves. (nature)
In an economic context, 'Zweig' (branch) is very often the second part of a compound noun. The most common examples are 'der Wirtschaftszweig' (economic sector) and 'der Industriezweig' (industry / branch of industry).
Der Tourismus ist für viele Regionen der wichtigste Wirtschaftszweig.
Tourism is the most important economic sector for many regions.
In the figurative meaning for subfields, 'Zweig' (branch) is often used with a genitive construction: 'ein Zweig der Forschung' (a branch of research), 'ein Zweig der Literatur' (a branch of literature), or 'ein Zweig der Familie' (a branch of the family).
Er forscht in einem relativ neuen Zweig der Physik.
He conducts research in a relatively new branch of physics.
arm
Der Arm and der Zweig can both refer to a part of a larger organization or structure. Zweig is usually the neutral term for this, while Arm sounds more figurative and presents the organization as a supporting or reaching-out unit. Zweig is more suitable in factual descriptions; Arm can also be used in more figurative phrasing.

Der kulturelle Zweig des Vereins organisiert Veranstaltungen für Jugendliche.
The association’s cultural branch organizes events for young people.
Der kulturelle Arm des Vereins organisiert Veranstaltungen für Jugendliche.
The association’s cultural arm organizes events for young people.