  The quantity that captures the asymptotic value of the maximum number of
appearances of a given topological tree (a rooted tree with no vertices of
outdegree $1$) $S$ with $k$ leaves in an arbitrary tree with sufficiently large
number of leaves is called the inducibility of $S$. Its precise value is known
only for some specific families of trees, most of them exhibiting a symmetrical
configuration. In an attempt to answer a recent question posed by Czabarka,
Sz\'ekely, and the second author of this article, we provide bounds for the
inducibility $J(A_5)$ of the $5$-leaf binary tree $A_5$ whose branches are a
single leaf and the complete binary tree of height $2$. It was indicated before
that $J(A_5)$ appears to be `close' to $1/4$. We can make this precise by
showing that $0.24707\ldots \leq J(A_5) \leq 0.24745\ldots$. Furthermore, we
also consider the problem of determining the inducibility of the tree $Q_4$,
which is the only tree among $4$-leaf topological trees for which the
inducibility is unknown.
