  Given $n$ points in the plane, a \emph{covering path} is a polygonal path
that visits all the points. If no three points are collinear, every covering
path requires at least $n/2$ segments, and $n-1$ straight line segments
obviously suffice even if the covering path is required to be noncrossing. We
show that every set of $n$ points in the plane admits a (possibly self-crossi
ng) covering path consisting of $n/2 +O(n/\log{n})$ straight line segments. If
the path is required to be noncrossing, we prove that $(1-\eps)n$ straight line
segments suffice for a small constant $\eps>0$, and we exhibit $n$-element
point sets that require at least $5n/9 -O(1)$ segments in every such path.
Further, the analogous question for noncrossing \emph{covering trees} is
considered and similar bounds are obtained. Finally, it is shown that computing
a noncrossing covering path for $n$ points in the plane requires $\Omega(n
\log{n})$ time in the worst case.
