Transversals in Trees
Victor A. Campos, Vašek Chvátal, Luc Devroye, Perouz Taslakian · Journal of Graph Theory · 2012
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 oftransversals 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 leastone 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 atmost d children has aunique minimal element with respect to this partial order and we describethis tree. In addition, we determine asymptotically the expected values of c(T,k) in special families of trees. © 2012 Wiley Periodicals, Inc.