The Maximum Number of Squares in a Tree

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 $\textsf{sq}(n)$ be the maximum number of different square substrings in a tree of size n. We show that asymptotically $\textsf{sq}(n)$ is strictly between linear and quadratic orders, for some constants c1,c2>0 we obtain: $$c_1n^{4/3} \le \textsf{sq}(n) \le c_2n^{4/3}.$$