We give a counterexample to the conjecture of Martin and Thatte that two balanced rooted binary leaf-labeled trees on $n$ leaves have a maximum agreement subtree (MAST) of size at least $n^{\frac{1}{2}}$. In particular, we show that for any $c>0$, there exist two balanced rooted binary leaf-labeled trees on $n$ leaves such that any MAST for these two trees has size less than $c n^{\frac{1}{2}}$. We also improve the lower bound of the size of such a MAST to $n^{\frac{1}{6}}$.

Phylogenetic Diversity (PD) is a prominent quantitative measure of the biodiversity of a collection of present-day species (taxa). This measure is based on the evolutionary distance among the species in the collection. Loosely speaking, if T is a rooted phylogenetic tree whose leaf set X represents a set of species and whose edges have real-valued lengths (weights), then the PD score of a subset S of X is the sum of the weights of the edges of the minimal subtree of T connecting the species in S. In this paper, we define several natural variants of the PD score for a subset of taxa which are related by a known rooted phylogenetic network. Under these variants, we explore, for a positive integer k, the computational complexity of determining the maximum PD score over all subsets of taxa of size k when the input is restricted to different classes of rooted phylogenetic networks.