Question:what is the probability that four distinct points in general position in the Euclidean plane are in convex configuration, depending on the number of leaf nodes in their Minimum Spanning Tree (MST)?

Put differently, what are the probabilities for the following situations:

- the points are in convex configuration and

- the MST has two leaf nodes?
- the MST has three leaf nodes?
- the points are in deltoid configuration

- the MST has two leaf nodes?
- the MST has three leaf nodes?

I assume that a quadruplet of points has a higher probability of being in convex configuration if its MST has two leaf nodes than if it has three, but that that probability is less than 1.

**Addendum:**

In reply to @GabeK's request for a definition of the sampling space, I would suggest the following, which deviates from what is the basis of answers to Sylvester's Four Point problem.

Instead of sampling the four points uniformly from planar geometric shape, whose set of inner points is path-connected, I request that the smallest closed disk that contains the four points is the unit disk and, that the sequence $(A,B,C,D)$ of points returned by the sampling process has the property, that $\lbrace A,B\rbrace$ defines one of the most distant pair of points and that the circum circle of $\lbrace A,B,C\rbrace$ isn't larger than the one of $\lbrace A,B,D\rbrace$.

The motivation for those restrictions on the sampling process is as follows:

- the probability of four points being in convex configuration should be independent of the order in which they are sampled
- the probabilites should be independent of similarity-preserving transformations of the Eulidean plane; i.e. if we calculate the probability that the points are in convex configuration before and after they have been subjected to such a transformation, both values should be the same because those transformations preserve convexity.
- we can always find a similarity-transformation, that takes the smallest enclosing circle of a quadruplet of points to the boundary of the unit disk.

All in all every quadruplet of points, whose smallest enclosing circle is the unit circle, represents transfinitely quadruplets of points that have been sampled from the entire Euclidean plane and have the same convexity predicate.