Further results on the inducibility of d-ary trees

Abstract

A subset of leaves of a rooted tree induces a new tree in a natural way. The density of a tree D inside a larger tree T is the proportion of such leaf-induced subtrees in T that are isomorphic to D among all those with the same number of leaves as D. The inducibility of D measures how large this density can be as the size of T tends to infinity. In this paper, we explicitly determine the inducibility in some previously unknown cases and find general upper and lower bounds, in particular in the case where D is balanced, i.e., when its branches have at least almost the same size. Moreover, we prove a result on the speed of convergence of the maximum density of D in strictly d-ary trees T (trees where every internal vertex has precisely d children) of a given size n to the inducibility as n → ∞, which supports an open conjecture.