Leaf realization problem, caterpillar graphs and prefix normal words

Given a simple graph $G$ with $n$ vertices and a natural number $i\le n$, 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 $({\mathrm{\ell }}_0,{\mathrm{\ell }}_1,\dots ,{\mathrm{\ell }}_n)$, there exists a simple graph $G$ with $n$ vertices such that ${\mathrm{\ell }}_i=L_G(i)$ for $i=0,1,\dots ,n$. 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 $(\mathrm{\Delta }{L}_G(i){)}_{1\le i\le n-3}$ and the set of prefix normal words.