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?