Maximizing the maximum degree in ordered nearest neighbor graphs

For an ordered point set in a Euclidean space or, more generally, in an abstract metric space, the ordered Nearest Neighbor Graph is obtained by connecting each of the points to its closest predecessor by a directed edge. We show that for every set of n points in $R^d$, there exists an order such that the corresponding ordered Nearest Neighbor Graph has maximum degree at least $\log n/\left(4d\right)$. Apart from the $1/\left(4d\right)$ factor, this bound is the best possible. As for the abstract setting, we show that for every n-element metric space, there exists an order such that the corresponding ordered Nearest Neighbor Graph has maximum degree $\Omega \left(\sqrt{\log n/\log \log n}\right)$.