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 induced subtrees.