(Elsevier, 2020) Czabarka, Eva; Dossou-Olory, Audace A. V.; Szekely, Laszlo A.; Wagner, Stephan

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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.

(Episciences, 2019) Dossou-Olory, Audace A. V.; Wagner, Stephan

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The quantity that captures the asymptotic value of the maximum number of appearances of a given topological tree (a rooted tree with no vertices of outdegree 1) S with k leaves in an arbitrary tree with sufficiently large number of leaves is called the inducibility of S. Its precise value is known only for some specific families of trees, most of them exhibiting a symmetrical configuration. In an attempt to answer a recent question posed by Czabarka, Sz´ekely, and the second author of this article, we provide bounds for the inducibility J(A5) of the 5-leaf binary tree A5 whose branches are a single leaf and the complete binary tree of height 2. It was indicated before that J(A5) appears to be ‘close’ to 1/4. We can make this precise by showing that 0.24707 . . . ≤ J(A5) ≤ 0.24745 . . .. Furthermore, we also consider the problem of determining the inducibility of the tree Q4, which is the only tree among 4-leaf topological trees for which the inducibility is unknown.