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I have attached an example of this structure for the word 'dream', from Blevins' chapter in the Handbook of Phonological Theory, 1995.

A branching syllable tree for the word 'dream'

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    Bolinger might be said to have prefigured in 1950 the [dr [i m]] structure with his notion of assonance and rime in phonosemantics. He wasn't interested in taking rimes apart further, though; that was considered obvious.
    – jlawler
    Jul 24, 2019 at 18:07

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The answer depends on what you mean by "hierarchical branching structure". The particular expression "hierarchical branching structure" was, AFAIK, first introduced in the late 70's in metrical studies, but the equivalent concepts are ancient. Candidates are "Ancient Chinese Grammarians", "Indic Grammarians" and possibly "Ancient Greek Grammarians".

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The structure is pretty clearly based off Chomskyan tree structures, so I can hardly imagine it being much earlier than that. Syntactic Structures is from 1957, so while the 60s would have been my guess, the 70s suggested in @user6726 's answer can't too far either.

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