  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.
