Let $\Gamma$ be a group generated by symmetric finite set $S$ and acting on $X$. *The Schreier graph* of the action is the graph with vertex set $X$ and $(x,y)$ is an edge if there is $s\in S$ such that $x=sy$.

Does there exists a faithful transitive action of the Thompson group F on a discrete set $X$ such that the Schreier graph of this action do not contain a binary tree?