Suppose we have a locally injective meromorphic function $f$ on some domain $D$. The three interesting cases for $D$ are the plane, the disc and an arbitrary domain. Suppose its poles are $P = \lbrace p_1, \dots, p_n \rbrace$ (assumed finite).
Question: Is there a tree contained in $D$ with endpoints $P$ on which $f$ is injective, aside from (possibly) the poles themselves?
If $D$ is a disk, it is not true, see the picture ($f$ exists by the Riemann mapping theorem). Not sure about the case when $D$ is the entire plane.