  The Markov triples, that means the positive integer solutions of the equation
$x^2+y^2+z^2=3xyz$, form the vertex set of the Markov tree. Each Markov triple
defines a weighted projective plane, which gives a geometric interpretation of
the vertex. We exhibit a class of rational, projective $\mathbb{C}^*$-surfaces
representing the edges of the Markov tree in the sense that they admit
coverings onto the adjacent weighted projective planes. Our main result shows
that the surfaces representing the Markov tree are precisely the normal
degenerations of the projective plane admitting a non-trivial
$\mathbb{C}^*$-action.
