  Given a simple graph $G$ with $n$ vertices and a natural number $i \leq n$,
let $L_G(i)$ be the maximum number of leaves that can be realized by an induced
subtree $T$ of $G$ with $i$ vertices. We introduce a problem that we call the
\emph{leaf realization problem}, which consists in deciding whether, for a
given sequence of $n+1$ natural numbers $(\ell_0, \ell_1, \ldots, \ell_n)$,
there exists a simple graph $G$ with $n$ vertices such that $\ell_i = L_G(i)$
for $i = 0, 1, \ldots, n$. We present basic observations on the structure of
these sequences for general graphs and trees. In the particular case where $G$
is a caterpillar graph, we exhibit a bijection between the set of the discrete
derivatives of the form $(\Delta L_G(i))_{1 \leq i \leq n - 3}$ and the set of
prefix normal words.

Combinatorics