Inducibility and universality for trees

Abstract

We answer three questions posed by Bubeck and Linial on the limit densities of subtrees in trees. We prove there exist positive ε 1 and ε 2 such that every tree that is neither a path nor a star has inducibility at most 1 − ε 1 , where the inducibility of a tree T is defined as the maximum limit density of T , and that there are infinitely many trees with inducibility at least ε 2 . Finally, we construct a universal sequence of trees; that is, a sequence in which the limit density of any tree is positive.