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Behavior of subtree of $\mathbb{Z}^2$ embedded in $\mathbb{C}$ under compactification of the latter to the riemann sphere

I consider a countable subtree $T$ of the integer lattice isomorphic to $\mathbb{Z}^2$ with directed edges. It shall be embedded in $\mathbb{C}$ where the edge $(u,v)$ points from $u$ to $v$ if and ...