finite but omnidirectional, and such that every vertex has finite degree. Assume further that for some vertex $\v$, $\calT$ is $\delta$-straight at $\v$. Then $\calT$ satisfies the following:

1.  Every semi-infinite path in $\calT$ starting from $\v$ has an asymptotic orientation;\

2.  For every $\alpha\in[0,2\pi)$ there exist at least one semi-infinite path in $\calT$ starting at $\v$ and with asymptotic orientation $e^{i\alpha}$.

3.  Every semi-infinite path $(\v_n)_{n\geq 1}$ in $\calT$ starting from $\v$ is $\delta$-straight about its asymptotic orientation $e^{i\alpha}$, i.e. $ang(\v_n,e^{i\alpha})<c|\v_n|^{-\delta}$ eventually.

Semi-infinite geodesics and the Busemann function {#geo}
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Semi-infinite geodesics: existence
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Recall that a path $\rho=(\v_1,\v_2,\dots)$ in $\calD$ is a semi-infinite geodesic if for all $\v_j,\v_k\in\rho$, the path $(\v_j,\v_{j+1},...,\v_k)$ is the unique geodesic connecting $\v_j$ to $\v_k$. Semi-infinite geodesics starting from $\v\in\calD_v$ and with asymptotic orientation $e^{i\alpha}$ are denoted $\rho_{\v}(\alpha)$.

\[pexis\] Let $\Omega_2$ be the event that for all semi-infinite geodesic $\rho$ there exists $\alpha=\alpha(\rho )\in [0,2\pi)$ such that $\rho$ is a $\alpha$-path, and that for all $\alpha\in[0,2\pi)$ and for all $\v\in\calD_v$ there exists at least one geodesic starting from $\v$ and with asymptotic orientation $e^{i\alpha}$. Under , $\bP(\Omega_2)=1$.

The first step to show the existence of semi-infinite geodesics and its convergence is the following result 