Given $n$ points in the plane, can we always find a cycle through all of them that has only straight line edges and no edges intersect (planar-drawn)?

Intuitively the answer is yes, but I am struggling with a proof. In the example above, the $2^{nd}$ graph demonstrates an incorrect algorithm, whereas the last graph demonstrates a working algorithm.

For different variants there are constraints - this is not always possible if the points are colored (e.g. if two points of one color lies on non-adjacent points of the convex hull and any path between them partitions the other color).

What if the graph is required to be a tree?