Justification of Sackin-index extensions as balance indices for phylogenetic networks

Determine whether the Sackin index extended to phylogenetic networks by defining each leaf’s depth as either (i) the length of a shortest directed path from the root or (ii) the length of a longest directed path from the root genuinely measures phylogenetic balance and therefore merits classification as a balance index for networks.

Background

The Sackin index is a classical tree balance index defined as the sum of leaf depths (distance from the root) on a phylogenetic tree. Extending this index to networks requires specifying what ‘depth’ means because multiple root-to-leaf paths may exist. Two natural choices are to use the shortest-path or the longest-path length as the depth of each leaf.

While these definitions are mathematically straightforward, the paper highlights uncertainty about whether they capture the intuitive notion of ‘balance’ for networks, in the same way the Sackin index is used for trees. Clarifying this would align network indices with the extensive literature on tree balance and help ensure interpretability in applications.