  In analogy to other concepts of a similar nature, we define the inducibility
of a rooted binary tree. Given a fixed rooted binary tree $B$ with $k$ leaves,
we let $\gamma(B,T)$ be the proportion of all subsets of $k$ leaves in $T$ that
induce a tree isomorphic to $B$. The inducibility of $B$ is $\limsup_{|T| \to
\infty} \gamma(B,T)$. We determine the inducibility in some special cases, show
that every binary tree has positive inducibility and prove that caterpillars
are the only binary trees with inducibility $1$. We also formulate some open
problems and conjectures on the inducibility. Finally, we present an
application to crossing numbers of random tanglegrams.
