Inducibility of d-ary trees

Abstract

Imitating the binary inducibility, a recently introduced invariant of binary trees (Cz- abarka et al., 2017), we initiate the study of the inducibility of d-ary trees (rooted trees whose vertex outdegrees are bounded from above by d ≥ 2). We determine the exact inducibility for stars and binary caterpillars. For T in the family of strictly d-ary trees (every vertex has 0 or d children), we prove that the difference between the maximum density of a d-ary tree D in T and the inducibility of D is of order O(|T |−1/2) compared to the general case where it is shown that the difference is O(|T |−1) which, in particular, responds positively to a conjecture on the inducibility in binary trees. We also discover that the inducibility of a binary tree in d-ary trees is independent of d. Furthermore, we establish a general lower bound on the inducibility and also provide a bound for some special trees. Moreover, we find that the maximum inducibility is attained for binary caterpillars for every d.