A height-based metaconcept for rooted tree balance and its implications for the $B_1$ index

TL;DR

This work introduces the height metaconcept (HM), a new height-based framework for assessing rooted tree balance, and proves that HM yields an imbalance index for all strictly increasing and 1-positive functions $f$, and a binary imbalance index for strictly increasing $f$. The authors fully characterize extremal trees: the caterpillar maximizes HM, while the greedily from the bottom tree (gfb-tree) minimizes HM, with all HM-minimizers sharing the gfb-tree's height sequence and height $h(T)=\lceil\log_2(n)\rceil$. They connect HM to the longstanding $B_1$ index, showing HM-minimizers coincide with $B_1$-maximizers and resolving two open Fischer2023 questions about maximum values and uniqueness of maximizers. Beyond $B_1$, the paper introduces three HM-derived indices including a root-inclusive variant, establishes their recursive structure, and derives their extremal values, linking one of them to an OEIS sequence. Overall, the height-based metaconcept broadens the toolkit for tree balance analysis and enables new, quantitatively distinct imbalance indices with clear extremal trees and values.

Abstract

Tree balance has received considerable attention in recent years, both in phylogenetics and in other areas. Numerous (im)balance indices have been proposed to quantify the (im)balance of rooted trees. A recent comprehensive survey summarized this literature and showed that many existing indices are based on similar underlying principles. To unify these approaches, three general metaconcepts were introduced, providing a framework to classify, analyze, and extend imbalance indices. In this context, a metaconcept is a function $Φ_f$ that depends on another function $f$ capturing some aspect of tree shape. In this manuscript, we extend this line of research by introducing a new metaconcept based on the heights of the pending subtrees of all inner vertices. We provide a thorough analysis of this metaconcept and use it to answer open questions concerning the well-known $B_1$ balance index. In particular, we characterize the tree shapes that maximize the $B_1$ index in two cases: (i) arbitrary rooted trees and (ii) binary rooted trees. For both cases, we also determine the corresponding maximum values of the index. Finally, while the $B_1$ index is induced by a so-called third-order metaconcept, we explicitly introduce three new (im)balance indices derived from the first- and second-order height metaconcepts, respectively, thereby demonstrating that pending subtree heights give rise to a variety of novel (im)balance indices.

Figures (14)

• Figure 1: Rooted binary tree $T$ with eight leaves and root $\rho$ (figure adapted from Fischer2025). Vertices $\rho$, $u$, and $v$ are ancestors of $v$. The parent of $v$ is $u$, and $v$ is one of two children of $u$. The descendants of $v$ are $v$, $z$, $w$, $x$, and $y$. The lowest common ancestor of $x$ and $z$ is $LCA_T(x,z) = v$. The leaves $x$ and $y$ form the cherry $[x,y]$, whose parent is $w$. The pending subtree of $u$ is $T_u$, which is also one of the two maximal pending subtrees of $T$. It has five leaves and thus $n_T(u) = 5$, i.e., the clade size of $u$ is five. The balance value of $v$ is one, i.e., $b_T(v) = 2-1 = 1$, hence $v$ is balanced. Vertices $u$ and $w$ are balanced, too, because $b_T(u) = 3-2 = 1$ and $b_T(w) = 1-1 = 0$. The root $\rho$ is not balanced as $b_T(\rho) = 5-3 = 2$.
• Figure 2: Examples of the special trees considered throughout this manuscript (figure adapted from Fischer2025). Note that the gfb-tree and the echelon tree generally do not coincide. For example, for $n = 5$, we have $T^{gfb}_5 = \left(T^{cat}_3,T^{cat}_2\right) \neq \left(T^{fb}_2,T^{cat}_1\right) = T^{be}_5$ (see Figure \ref{['Fig:5_2_3']} in the appendix).
• Figure 3: For $n = 9$, there are five trees, namely $T_1, T_2, T_3, T^{be}_9$ and $T^{gfb}_9$, in $\mathcal{BT}^{\ast}_9$ that minimize the HM. In particular, they all share the height sequence $\mathcal{H}(T) = (1,1,1,1,2,2,3,4)$.
• Figure 4: Example of trees $T$ and $T'$ as described in the proof of Lemma \ref{['Lem:H_seq_relocate_cherry_to_cherry']}. Here, $k=8$, $j=6$, and $l=4$, and the vertices are grouped into six groups $G_1, \ldots, G_6$.
• Figure 5: $T$ and $T'$ as described in the proof of Lemma \ref{['Lem:H_seq_caterpillar']} and Theorem \ref{['Theo:B1_extrema']}.

Theorems & Definitions (68)

• Remark 2.1
• Definition 2.2: (Binary) (im)balance index (Fischer2023)
• Definition 2.3: Locality (Mir2013Fischer2023)
• Definition 2.4: Recursiveness (based on Matsen2007Fischer2023)
• Lemma 2.5: Fischer2025, Lemma 3.4
• Proposition 2.6: Coronado2020a, Proposition 5
• Remark 3.2: adapted from Fischer2025, Remark 3.2
• Remark 3.3
• Theorem 3.4
• Theorem 3.5