Table 4.

This table provides an overview of our results, indicating for which families of the function f the metaconcepts qualify as (binary) imbalance indices. The column labels refer to four cases, all of which require f to be strictly increasing. A checkmark ($✓$) indicates that no further conditions on f are needed to satisfy the corresponding property. When additional constraints on f are required, they are explicitly stated in the respective cell. Conversely, a cross ($\times$) indicates that for at least one function in the given family, the metaconcept fails to be a (binary) imbalance index. The entry “depends” means that the binary minimizing tree(s) are not the same for all functions within that family. An entry in square brackets indicates that this result is adapted from Cleary et al. (2025), Corollary 4.4

			f strictly increasing and
			–	convex	str. concave	affine ($m>0$, $a\in \mathbb{R}$)
${\mathrm{\Phi }}_f^{\mathcal{B}}$	imb. index on $\mathcal{B}\mathcal{T}_n^{\ast }$		$✓$ (Theo. 3.6)	$✓$ (Theo. 3.6)	$✓$ (Theo. 3.6)	$✓$, $\equiv C(T)$ (Rem. 3.5)
	min. tree(s) on $\mathcal{B}\mathcal{T}_n^{\ast }$		depends (Rem. 3.3)	e.g., $T_n^{\mathit{mb}}$ (Theo. 3.8)	depends (Rem. 3.3)	$argminC(T)$ (Cor. 3.7)
${\mathrm{\Phi }}_f^{\mathcal{N}}$	imb. index on	$\mathcal{B}\mathcal{T}_n^{\ast }$	$\times$ (Rem. 3.3)	str. convex: $✓$ (Cor. 3.20)	[$✓$] (Prop. 2.10)	$✓$, $\equiv S(T)$ (Rem. 3.14)
	imb. index on	$\mathcal{T}_n^{\ast }$	$\times$ (Rem. 3.3)	str. convex, 2-positive: $✓$ (Cor. 3.20)	2-positive: $✓$ (Cor. 3.16)	$a\ge 0$: $✓$, $a=0\Rightarrow \equiv S(T)$ (Rem. 3.14, Prop. 3.21)
	min. tree(s) on $\mathcal{B}\mathcal{T}_n^{\ast }$		depends (Theo. 3.17, Prop. 2.10)	str. convex: $T_n^{\mathit{mb}}$ (Theo. 3.17)	$\left[T_n^{\mathit{gfb}}\right]$ (Prop. 2.10)	$argminS(T)$ (Prop. 3.21)
${\mathrm{\Phi }}_f^{\mathrm{\Delta }}$	imb. index on	$\mathcal{B}\mathcal{T}_n^{\ast }$	$\times$ (Rem. 3.3)	$✓$ (Prop. 3.30)	$\times$ (Rem. 3.3)	$✓$, $\equiv S(T)$ (Rem. 3.29)
	imb. index on	$\mathcal{T}_n^{\ast }$	$\times$ (Rem. 3.3)	$✓$ (Prop. 3.30)	$\times$ (Rem. 3.3)	$✓$, $\equiv S(T)$ (Rem. 3.29)
	min. tree(s) on $\mathcal{B}\mathcal{T}_n^{\ast }$		depends (Rem. 3.3)	$argminS(T)$ (Prop. 3.30)	depends (Rem. 3.3)	$argminS(T)$ (Prop. 3.31)