Leaf realization problem, caterpillar graphs and prefix normal words

Given a simple graph G with n vertices and a natural number i 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 leaf realization problem, which consists in deciding whether, for a given sequence of n + 1 natural numbers (l(0), l(1),...,l(n)), there exists a simple graph G with n vertices such that l(i) = L-G(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 (Delta L-G(i))(1 <= i <= n-3) and the set of prefix normal words. (C) 2018 Elsevier B.V. All rights reserved.