Sackin indices for labeled and unlabeled classes of galled trees

The Sackin index is an important measure for the balance of phylogenetic trees. We investigate two extensions of the Sackin index to the class of galled trees and two of its subclasses, namely simplex galled trees and normal galled trees, where for all classes we consider both labeled and unlabeled galled trees. In all cases, we show that the mean of the Sackin index for a network which is uniformly sampled from its class is asymptotic to $\mu n^{3/2}$ for an explicit constant $\mu$ . In addition, we show that the scaled Sackin index converges weakly and with all its moments to the Airy distribution.