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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 inducedsubtree $T$ of $G$ with $i$ vertices. We introduce a problem that we call the\emph{leaf realization problem}, which consists in deciding whether, for agiven 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 ofthese 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 discretederivatives of the form $(\Delta L_G(i))_{1 \leq i \leq n - 3}$ and the set ofprefix normal words.

Leaf realization problem, caterpillar graphs and prefix normal words