Decks of rooted binary trees

We consider extremal problems related to decks and multidecks of rooted binary trees (a.k.a. rooted phylogenetic tree shapes). Here, the deck (resp. multideck) of a tree T refers to the set (resp. multiset) of leaf-induced binary subtrees of T. On the one hand, we consider the reconstruction of trees from their (multi)decks. We give lower and upper bounds on the minimum (multi)deck size required to uniquely encode a rooted binary tree on n leaves. On the other hand, we consider problems related to deck cardinalities. In particular, we characterize trees with minimum-size as well as maximum-size decks. Finally, we present some exhaustive computations for k-universal trees, i.e., rooted binary trees that contain all k-leaf rooted binary trees as leaf-induced subtrees.