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It's well known that the $A_{\infty}$ and $L_{\infty}$ operads, being resolutions of the associative and Lie operads, admit descriptions as free operads of certain trees.

The description I am referring to for the $A_{\infty}$ operad is described in Voronov's notes: http://www-users.math.umn.edu/~voronov/8390/lec9.pdf

Similarly the $L_{\infty}$ operad is described in section 2 of the paper by Voronov, Stasheff and Kimura: https://arxiv.org/pdf/hep-th/9307114.pdf

Does the $G_{\infty}$-operad, the free resolution of the Gerstenhaber (Poisson) operad admit a similar description in terms of trees? How can one define it explicitly?

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Yes. All of your examples fall within the techniques of Koszul duality theory for operads, and for this you can consult the standard reference of J.-L. Loday and B. Vallette. But also see G. Ginot's thesis http://www.numdam.org/item/AMBP_2004__11_1_95_0/ for explicit descriptions of $\mathsf{Gers}_\infty$ and $\mathsf{Pois}_\infty$.

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