  A transversal in a rooted tree is any set of nodes that meets every path from
the root to a leaf. We let c(T,k) denote the number of transversals of size k
in a rooted tree T. We define a partial order on the set of all rooted trees
with n nodes by saying that a tree T succeeds a tree T' if c(T,k) is at least
c(T',k) for all k and strictly greater than c(T',k) for at least one k. We
prove that, for every choice of positive integers d and n, the set of all
rooted trees on n nodes where each node has at most d children has a unique
minimal element with respect to this partial order and we describe this tree.
