which is delimited by the rightmost path $R_S$ of $S$ from the left and by $h^1_{\Gamma}$ and $h^2_{\Gamma}$ from above and below, respectively. Then no node or edge of $S$, other than those of $R_S$, intersects $\mathcal R_{\Gamma}$.

We first argue about possible intersections with the interior of $\mathcal R_{\Gamma}$.

-   Suppose, for a contradiction, that an edge of $S$ intersects the interior of $\mathcal R_{\Gamma}$. Since $S$ is connected, it follows that there is an edge $e$ of $S$ that intersects the interior of $\mathcal R_{\Gamma}$ [*and*]{} its boundary. Since $\Gamma$ is planar, the edge $e$ does not cross $R_S$. Since $R_S$ is the rightmost path of $S$, the edge $e$ does not share a node with $R_S$. Since $\Gamma$ is strictly-upward and $r(S)$ is the root of $S$, we have that $e$ does not intersect $h^1_{\Gamma}$ other than, possibly, at $r(S)$; however, $r(S)$ belongs to $R_S$, and we already ruled out the possibility that $e$ shares a node with $R_S$. Finally, if $e$ intersects $\mathcal R_{\Gamma}$ and $h^2_{\Gamma}$, then, since $\Gamma$ is strictly-upward, the path from $e$ to $r(S)$ intersects $R_S$, hence there is an edge $e'$ of $S$ that intersects $\mathcal R_{\Gamma}$ and $R_S$, a case which 