Some facts about cut-locus Let $M$ be a 2-dimensional closed Riemmanian manifold diffeomorphic  to $S^2$.
S.B.Myers says "the cut-locus of every point $x\in M$ is a finite tree."

  
*
  
*How the set of point  can be a tree? What are the edges?
  
*Is $p$ an element of $\operatorname{cut-locus}(p)$?
  

I can not find any paper of Myers in this case.
Thanks!
 A: I think Myers only considered analytic metrics, see his papers "Connections between differential geometry and topology I and II", Duke Math. J. 1 (1935), 376-391, and 2 (1936), 95-102. 
For arbitrary metrics on $S^2$ the cut locus is indeed a tree. This can be deduced from e.g. in [Shiohama and Tanaka, Cut loci and distance spheres on Alexandrov surfaces] who work in the (more general) settings of Alexandrov spaces homeomorphic to surfaces and prove that the cut locus is a local tree.
(This paper is available online, search by title). Specializing to the case when the surface is a Riemannian sphere we observe:


*

*The complement to the cut locus is a 2-disk. 

*If the cut locus contains an embedded circle, then by Jordan curve theorem the circle separates $S^2$ into two disks, and by part 1 one of the disks must lie in the cut locus, so the cut locus cannot be  local tree.
One should be careful with what is meant by a local tree. By a result of Gluck and Singer [Scattering of Geodesic Fields II, Annals of Mathematics, Second Series, Vol. 110, No. 2 (Sep., 1979), pp. 205-225] there is a positively curved Riemannian metric on $S^2$, in fact a convex surface of revolution, for which the cut locus cannot be triangulated, so it is definitely not a finite tree. 
For recent works in this area see papers of Itoh, e.g. http://arxiv.org/pdf/1103.1758 and references therein. 
A: If you will excuse me substituting a polyhedron for the Riemannian manifold
(imagine rounding the vertices),
this figure shows how the cut locus (red) from source $x$ is a tree.
(The green arcs are equidistant from $x$.)
     
     (Figure from Discrete and Computational Geometry)

Concerning, "What are the edges?": They are geodesics, such that there are two
distinct shortest paths from $x$ to every interior point of an edge of the cut locus.
Points of the cut locus of degree $k$ have $k$ distinct shortest paths from $x$.
