  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 switches on
a binary tree. For a given binary tree, we formalize the constraints on the
coloring, in particular the distribution of the nodes among colors. Following
Guidon, we are interested in balanced colorings i.e. colorings which minimize
the maximum size of the subsets of the tree nodes distributed by color. With
his method, we present balanced colorings for trees of height up to 7. But his
method seems difficult to apply for trees of greater height. Also we present
another method which gives solutions for arbitrarily large trees. We illustrate
it with a balanced coloring for height 8. In the appendix, we give the exact
formulas and the asymptotic behavior of the number of colorings as a function
of the height of the tree.
