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A subset of leaves of a rooted tree induces a new tree in a natural way. Thedensity of a tree $D$ inside a larger tree $T$ is the proportion of suchleaf-induced subtrees in $T$ that are isomorphic to $D$ among all those withthe same number of leaves as $D$. The inducibility of $D$ measures how largethis density can be as the size of $T$ tends to infinity. In this paper, weexplicitly determine the inducibility in some previously unknown cases and findgeneral upper and lower bounds, in particular in the case where $D$ isbalanced, 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 $D$ instrictly $d$-ary trees $T$ (trees where every internal vertex has precisely $d$children) of a given size $n$ to the inducibility as $n \to \infty$, whichsupports an open conjecture.

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