Sufficiency of the inequalities (1–5) for perfect binary trees

Prove that, for any perfect binary tree T of height h, every non-negative integer sequence A = (a1, a2, ..., ac) with c ≥ h + 1 that satisfies the necessary constraints relating A to the node counts by height (n0, n1, ..., nh)—namely (i) sum_{i=1}^c a_i = n (the total number of nodes of T), (ii) a_j = 1 for at least one color j (the root’s color), and (iii) for every subset of k distinct colors (1 ≤ k ≤ c) the sum of their counts is at most n0 + n1 + ... + n_{k-1} where n_g denotes the number of nodes of height g—can be realized as a valid coloring of T under the rule that no node shares a color with any ancestor or descendant (i.e., that these necessary conditions are also sufficient for perfect binary trees).

Background

The paper studies a coloring rule on rooted trees motivated by a remote control system for a rail yard: no node may share a color with any ancestor or descendant. For a coloring using c colors, the authors define a colorable partition A = (a1, a2, ..., ac) where ai counts the nodes colored i, and relate it to the sequence (n0, n1, ...) of numbers of nodes by height.

They prove that A must satisfy a set of inequalities (Eqs. (1–5)) involving only (n0, n1, ...): the total count constraint, that at least one color appears exactly once (the root’s color), and that the sum of counts over any k colors is bounded by n0 + n1 + ... + n_{k-1}. These conditions are necessary for all rooted trees. They also exhibit rooted, binary, and full binary trees where these conditions are not sufficient due to structural imbalance.

Because perfect binary trees are maximally balanced (with n_g = 2^{h-g} nodes of height g), the authors conjecture that the necessary conditions become sufficient on this class, i.e., any sequence A satisfying (1–5) corresponds to an actual valid coloring of a perfect binary tree.