Further results on the inducibility of $d$-ary trees

A subset of leaves of a rooted tree induces a new tree in a natural way. The density of a tree $D$ inside a larger tree $T$ is the proportion of such leaf-induced subtrees in $T$ that are isomorphic to $D$ among all those with the same number of leaves as $D$. The inducibility of $D$ measures how large this density can be as the size of $T$ tends to infinity. In this paper, we explicitly determine the inducibility in some previously unknown cases and find general upper and lower bounds, in particular in the case where $D$ is balanced, i.e., when its branches have at least almost the same size. Moreover, we prove a result on the speed of convergence of the maximum density of $D$ in strictly $d$-ary trees $T$ (trees where every internal vertex has precisely $d$ children) of a given size $n$ to the inducibility as $n \to \infty$, which supports an open conjecture.