Inducibility of topological trees

Trees without vertices of degree 2 are sometimes named topological trees. In this work, we bring forward the study of the inducibility of (rooted) topological trees with a given number of leaves. The inducibility of a topological tree S is the limit superior of the proportion of all subsets of leaves of T that induce a copy of S as the size of T grows to infinity. In particular, this relaxes the degree-restriction for the existing notion of the inducibility in d-ary trees. We discuss some of the properties of this generalised concept and investigate its connection with the degree-restricted inducibility. In addition, we prove that stars and binary caterpillars are the only topological trees that have an inducibility of 1. We also find an explicit lower bound on the limit inferior of the proportion of all subsets of leaves of T that induce either a star or a binary caterpillar as the size of T tends to infinity.