Graphs and Combinatorics

, Volume 35, Issue 1, pp 303–320 | Cite as

The Minimum Asymptotic Density of Binary Caterpillars

  • Audace A. V. Dossou-OloryEmail author
Original Paper


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 


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Copyright information

© Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesStellenbosch UniversityMatielandSouth Africa

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