The Minimum Asymptotic Density of Binary Caterpillars

Abstract

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.