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

TL;DR

This work addresses coloring a perfect binary tree under an ancestor-different rule, seeking balanced colorings that minimize the largest color class among the $h+1$ colors required. It introduces a recursive algorithm that, given any colorable partition $A=(a_0, ldots,a_h)$ with $a_0=1$, constructs compatible partitions for the two subtrees and colors the entire tree, enabling balance for arbitrarily large heights. The approach generalizes Guidon's height-limited results by providing a constructive method that works beyond height $7$, including a detailed height-8 example, and analyzes the counting of colorings via the appendix. The methodology offers a practical tool for designing balanced — yet potentially fragmented — colorings and highlights a trade-off between balance and spatial color grouping, with implications for related remote-control or network-design problems where node-color distributions impact performance.

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.

Figures (7)

• Figure 1: Canonical coloring of a tree with height 3.
• Figure 2: The two non-equivalent colorings of a tree of height 2 with 3 colors, the canonical one (left) and another (right).
• Figure 3: The 3 colorings of a tree of height 3 with colorable partition $A = (1,2,6,6)$. For each coloring, there are one node with color 0, two nodes with color 1, six nodes with color 2 and six nodes with color 3.
• Figure 4: A balanced coloring $(1,4,5,5)$ given by Guidon Guidon-1 for a tree with height 3. The drawing on the right is a compact representation of the one on the left. For convenience, the colors are now labeled by letters $a,b,c,d$ and a notation like $b-2c-4d$ represents a subtree with a canonical coloring where each $2^k$ is the number of nodes with the same height and the same color: one node colored with $b$, 2 nodes with color $c$ and 4 nodes with color $d$.
• Figure 5: Left: a balanced coloring $(1,7,7,8,8)$ for a tree with height 4. Right: a balanced coloring $(1,12,12,12,13,13)$ for a tree with height 5. These two solutions are given by Guidon Guidon-1.