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All answers having "Branch" as clue:

3 Character Answers:

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ARM

RHO



4 Character Answers:

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LIMB

RAME

SECT

TWIG

UNIT

WING



5 Character Answers:

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BOUGH

OLIVE

PRONG

RAMUS

STICK



8 Character Answers:

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OFFSHOOT



9 Character Answers:

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AFFILIATE