Remote control system of a binary tree of switches -- I. constraints and inequalities

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 $i$.