  Let $\mathcal{T}$ be an unrooted binary tree with $n$ distinctly labelled
leaves. Deriving its name from the field of phylogenetics, a convex character
on $\mathcal{T}$ is simply a partition of the leaves such that the minimal
spanning subtrees induced by the blocks of the partition are mutually disjoint.
In earlier work Kelk and Stamoulis (Advances in Applied Mathematics 84 (2017),
pp. 34--46) defined $g_k(\mathcal{T})$ as the number of convex characters where
each block has at least $k$ leaves. Exact expressions were given for $g_1$ and
$g_2$, where the topology of $\mathcal{T}$ turns out to be irrelevant, and it
was noted that for $k \geq 3$ topological neutrality no longer holds. In this
article, for every $k \geq 3$ we describe tree topologies achieving the maximum
and minimum values of $g_k$ and determine corresponding expressions and
exponential bounds for $g_k$. Finally, we reflect briefly on possible
algorithmic applications of these results.
