Table 2.

Definitions of binary imbalance indices, which are only applicable to binary trees, i.e., those with domain $\mathcal{B}\mathcal{T}_n^{\ast }$. It is straightforward to see that these binary imbalance indices are induced by the balance value metaconcept ${\mathrm{\Phi }}_f^{\mathcal{B}}$ when the function f is chosen as specified in the right column. The (quadratic) Colless index is induced by the first-order metaconcept where $\mathit{id}$ is the identity function, while the corrected Colless index is induced by the second-order metaconcept.

binary imbalance index	definition	BVM ${\mathrm{\Phi }}_f^{\mathcal{B}}$
Colless index Colless (1982); Shao and Sokal (1990)	$C\left(T\right):=\sum _{v\in \mathring{V}\left(T\right)}{b}_T(v)$	$\mathit{id}$
Corrected Colless index Heard (1992)	$I_C\left(T\right):=\frac{2}{(n-1)(n-2)}·C\left(T\right);$ $I_C\left(T\right):=0$ for $n=1,2$	$f_{I_C}(b,n)=\frac{2}{(n-1)(n-2)}·b$
Quadratic Colless index Bartoszek et al. (2021)	$QC\left(T\right):=\sum _{v\in \mathring{V}\left(T\right)}{b}_T{(v)}^2$	$f_{\mathit{QC}}(b)=b^2$