![Illustration of the convergence to a tree of spheres.[]{data-label="dmpp"}](dmpp-eps-converted-to.pdf){width="14cm"}

\[zoomstof\] When we consider the normalization $M_n\circ i_n$, we see that the images of $c$ and $d$ are collapsing so intuitively, considering the normalization $N_n\circ i_n$ corresponds to zooming or rescaling at the level of these two points.

Before developing the definition of trees of spheres in the next subsection, let us first describe this example. The tree of sphere on Figure \[dmpp\] has 5 edges, and 6 vertices. There are two categories of vertices: elements of $X$ at the ends (or leaves) of the tree and spheres at the interior and on which are attached edges. At each internal vertex $\S_v$ the set of branches induces a partition of $X$. Hence this naturally provides an application $X\to\S_v$ that maps each element of $x$ to the attaching point of the branch to which it belongs. The two spheres provide two applications that correspond exactly to the ones associated to $M_n$ and $N_n$. Note that from the definition, both of the spheres have at least three different marked points. Of course, in order to have a well define and unique limit we have to consider these trees modulo a natural action of 