Tight bound for the number of distinct palindromes in a tree

Abstract

For an undirected tree with n edges labelled by single letters, we consider its substrings, which are labels of the simple paths between pairs of nodes. We prove that there are O(n1.5) different palindromic substrings. This solves an open problem of Brlek, Lafrenière and Provençal (DLT 2015), who gave a matching lower-bound construction. Hence, we settle the tight bound of Θ(n1.5) for the maximum palindromic complexity of trees. For standard strings, i.e., for paths, the palindromic complexity is n + 1.