Leaf realization problem, caterpillar graphs and prefix normal words

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.