($\delta$-straightness). A sufficient condition for $\delta$-straightness is given by the next lemma, which is exactly Lemma 2.7 in [@hn01]:

\[hn-101\] If $(\x_n)_\bN$ is a sequence of points in $\bR^2$ with $|\x_n|\to\infty$ when $n\to\infty$, and such that for all large $n$ $$|\x_{n+1}-\x_n|\leq |\x_n|^{1-\delta}\mbox{ and }d(\x_n,[\x_1,\x_m])\leq |\x_m|^{1-\delta}\mbox{ for }m>n\,,$$ then there exist a contant $c>0$ such that for all $n$ sufficiently large, $$\label{eang}
ang(\x_n,\x_m)\leq c|\x_n|^{-\delta}\,\mbox{ whenever }m>n\,.$$

We also consider the $\delta$-straightness property for trees (we have the tree of infection in mind) as follows. For $\epsilon\in[0,\pi)$ let $$\C(\x,\epsilon):=\{\y\in\bR^{2}\backslash\{0\}\,:\,ang(\y,\x)\leq\epsilon\}\,.$$ If $\calT$ is a tree embedded in $\bR^{2}$, for each pair $\v,\tilde{\v}\in\calT$ let $\calR_{out}(v,\tilde{v})$ be the set of all $\hat{\v}\in\calT$ such that the unique path in $\calT$ connecting $\v$ to $\hat{\v}$ touches $\tilde{\v}$. For $\delta\in(0,1)$, define that $\calT$ is $\delta$-straight at $\v$ if, for all but finitely many $\tilde{\v}\in\calT$, $$\calR_{out}(\v,\tilde{\v})\subseteq \:\v+C(\tilde{\v}-\v,c|\tilde{\v}-\v|^{-\delta})\,.$$ We say that a subset $\calP$ of $\bR^2$ is omnidirectional if, for all $M>0$, the set composed of unit vectors $\v/|\v|$ with $\v\in\calP$ and $|\v|>M$ is dense in $\S^1$. The above lemma, which is Proposition 2.8 in [@hn01], states that $\delta$-straightness implies existence of an asymptotic orientation:

\[hn-201\] Assume that $\calT$ is a tree embedded in $\bR^2$, whose vertex set is locally 