Remote control system of a binary tree of switches -- II. balancing for a perfect binary tree

TL;DR

This paper investigates balanced coloring methods for binary trees, extending previous work to larger trees, and provides formulas and asymptotic analysis for the number of such colorings.

Contribution

It introduces a new method for constructing balanced colorings applicable to arbitrarily large binary trees, surpassing previous limitations.

Findings

Balanced colorings for trees of height up to 7 using existing methods.

A new method enabling balanced colorings for arbitrarily large trees, demonstrated on height 8.

Formulas and asymptotic behavior for the number of colorings as tree height increases.

Abstract

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.