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The translations I have seen of this translate it, "She gives it [referring to the oracular seat previously mentioned] to Phoebus as a birthday gift."
Doesn't there need to be a pronoun acting as direct object which refers back to the seat? Can you just insert the particle "as" without any textual justification?
well, to keep things simple, since it's easily inferred you don't need a toyton or anything there. For it to translate right however you need both "it" and "as"
Does this type of ellipsis occur frequently in Greek tragedy? I have just begun studying Greek poetry, and I still am not thoroughly acquainted with the license they take with the language.
I'm not very familiar with tragedy myself, but I suspect that this type of omission of the pronoun is much more common in tragedy than in prose or even epic poetry.
Regarding the "as" business, it's not really an omission here: that is a normal use of the accusative case. Hopefully someone will correct me if I'm wrong, but I think γενέθλιον δόσιν is in apposition (to the whole clause, I think, and not just the omitted pronoun) as an explanation of why it was given. And the accusative in apposition is often cognate: that is, notice that δόσιν contains the same idea as the verb, δίδωσι. It might be redundant to say "giving as a gift" (although Homer seems to do this, like "fighting a fight"), but here we get the added information that it is a birthday gift. The first example (although not cognate) that comes to mind is from the Hymn to Demeter: νά?κισσόν, ὃν φῦσε δόλον κού?ῃ, the narcissus, which she grew as a trap for the girl. The word δόλον, trick, trap, is appositive in order to explain why she (the Earth) grew the flower.
But, although I'm sure you'll see the acc used like this regularly, I'm not sure that my explanation is correct, so don't trust me too much.