  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.
