The maximum number of squares in a tree

Abstract

We show that the maximum number of different square substrings in unrooted labelled trees behaves much differently than in words. A substring in a tree corresponds (as its value) to a simple path. Let be the maximum number of different square substrings in a tree of size n. We show that asymptotically is strictly between linear and quadratic orders, for some constants c 1,c 2 > 0 we obtain: c 1n 4/3 ≤ sq(n) ≤ c 2n 4/3. © 2012 Springer-Verlag.