Given \(d\ge 2\) and two rooted d-ary trees D and T such that D has k leaves, the density \(\gamma (D,T)\) of D in T is the proportion of all k-element subsets of leaves of T that induce a tree isomorphic to D, after contracting all vertices of outdegree 1. In a recent work, it was proved that the limit inferior of this density as the size of T grows to infinity is always zero unless D is the k-leaf binary caterpillar \(F^2_k\) (the binary tree with the property that a path remains upon removal of all the k leaves). Our main theorem in this paper is an exact formula (involving both d and k) for the limit inferior of \(\gamma (F^2_k,T)\) as the size of T tends to infinity.
Caterpillars Minimum asymptotic density Leaf-induced subtrees d-ary trees Inducibility Complete d-ary trees Strict d-ary trees
Mathematics Subject Classification
Primary 05C05 Secondary 05C07 05C30 05C35
This is a preview of subscription content, log in to check access.
Czabarka, É., Dossou-Olory, A.A.V., Székely, L.A., Wagner, S.: Inducibility of \(d\)-ary trees (2018). arXiv:1802.03817
Czabarka, É., Székely, L.A., Wagner, S.: Inducibility in binary trees and crossings in random tanglegrams. SIAM J. Discrete Math. 31(3), 1732–1750 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
Dossou-Olory, A.A.V., Wagner, S.: Further results on the inducibility of \(d\)-ary trees. arXiv:1811.11235(to be submitted)
Fischermann, M., Hoffmann, A., Rautenbach, D., Székely, L., Volkmann, L.: Wiener index versus maximum degree in trees. J. Discrete Math. 122(1–3), 127–137 (2002)MathSciNetCrossRefzbMATHGoogle Scholar