  We study a tree coloring model introduced by Guidon (2018), initially based
on an analogy with a remote control system of a rail yard, seen as a switch
tree. For a given rooted tree, we formalize the constraints on the coloring, in
particular on the minimum number of colors, and on the distribution of the
nodes among colors. We show that the sequence $(a_1,a_2,a_3,\cdots)$, where
$a_i$ denotes the number of nodes with color $i$, satisfies a set of
inequalities which only involve the sequence $(n_0,n_1,n_2,\cdots)$ where $n_i$
denotes the number of nodes with height $i$. By coloring the nodes according to
their depth, we deduce that these inequalities also apply to the sequence
$(d_0,d_1,d_2,\cdots)$ where $d_i$ denotes the number of nodes with depth $i$.
