Table 1.

Definitions of imbalance indices that are applicable to arbitrary trees, i.e., those with domain $\mathcal{T}_n^{\ast }$ (notice that the $\widehat{s}$-shape statistic is applicable to arbitrary trees, but is only an imbalance index on $\mathcal{B}\mathcal{T}_n^{\ast }$). It is straightforward to see that these imbalance indices are induced by the clade size metaconcept ${\mathrm{\Phi }}_f^{\mathcal{N}}$ and the leaf depth metaconcept ${\mathrm{\Phi }}_f^{\mathrm{\Delta }}$, respectively, when the function f is chosen as specified in the two rightmost columns. The Sackin index and the $\widehat{s}$-shape statistic are induced by the first-order metaconcept with $\mathit{id}$ as the identity function. In contrast, the average leaf depth is induced by the second-order metaconcept. Moreover, the total cophenetic index is induced on $\mathcal{B}\mathcal{T}_n^{\ast }$ by the second-order and on $\mathcal{T}_n^{\ast }$ by the third-order metaconcept. This is because for binary trees we have $|\mathring{V}(T)|=n-1$, so no further additional value than n is needed. For further details on the total cophenetic index, see Remark 3.14

imbalance index	definition	CSM ${\mathrm{\Phi }}_f^{\mathcal{N}}$	LDM ${\mathrm{\Phi }}_f^{\mathrm{\Delta }}$
Sackin index Sackin (1972); Shao and Sokal (1990); Blum and François (2005); Fischer (2021)	$S(T):=\sum _{x\in V_L(T)}{\delta }_T(x)$ $\phantom{S(T)}=\sum _{v\in \mathring{V}(T)}{n}_T(v)$	$\mathit{id}$	$\mathit{id}$
Average leaf depth Kirkpatrick and Slatkin (1993)	$\overline{N}(T):=\frac{1}{n}\sum _{x\in V_L(T)}{\delta }_T(x)$ $\phantom{\overline{N}(T)}=\frac{1}{n}·S(T)$	$f_{\overline{N}}(n_v,n)=\frac{1}{n}·n_v$	$f_{\overline{N}}(\delta ,n)=\frac{1}{n}·\delta$
$\widehat{s}$-shape statistic Blum and François (2006)	$\widehat{s}(T):=\sum _{v\in \mathring{V}(T)}log\left(n_T,(v),-,1\right)$	$f_{\widehat{s}}(n_v)=log(n_v-1)$	–
Total cophenetic index Mir et al. (2013)	$\mathrm{\Phi }(T):=\sum _{\begin{array}{c}(x,y)\in V_L{(T)}^{2}x\ne y\end{array}}{\phi }_T(x,y)$ $\phantom{\mathrm{\Phi }(T)}=\sum _{v\in \mathring{V}(T)\backslash \left\{\rho \right\}}\left(\begin{array}{c}{n}_T(v)\\ 2\end{array}\right)$	$f_{\mathrm{\Phi }}(n_v,n,|\mathring{V}(T)|)=\left(\begin{array}{c}{n}_v\\ 2\end{array}\right)-\frac{\left(\begin{array}{c}n\\ 2\end{array}\right)}{|\mathring{V}(T)|}$	–