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 ( 0 , 1 , . . . , n ), there exists a simple graph G with n vertices such that 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 (∆L G (i)) 1≤i≤n−3 and the set of prefix normal words.