Exact expected s-shape statistics under standard tree distributions

Determine the exact expected value of the s-shape statistic for rooted binary trees under the uniform distribution and the YuleHarding distribution.

Background

The paper studies the $\widehat{s}$-shape statistic, defined as the sum of $\log(n_v-1)$ over internal vertices of a rooted binary tree, under two standard probabilistic models of phylogenetic trees: the uniform model and the YuleHarding model. These expectations are relevant for understanding the typical balance of random trees under the two distributions.

The authors state that the expected values under both models were previously unknown and then derive bounds for the uniform model rather than exact formulas. Consequently, determining exact expected values, particularly for the YuleHarding model and an exact uniform-model expression beyond the stated bounds, remains unresolved in the paper.