Leaf realization problem, caterpillar graphs and prefix normal words

Given a simple graph G with n vertices and a natural number i≤n, let LG(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 leaf realization problem, which consists in deciding whether, for a given sequence of n+1 natural numbers (ℓ0,ℓ1,…,ℓn), there exists a simple graph G with n vertices such that ℓi=LG(i) for i=0,1,…,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 (ΔLG(i))1≤i≤n−3 and the set of prefix normal words.