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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 $$\\mathcal {O}n^{1.5}$$ 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 $$\\Theta n^{1.5}$$ for the maximum palindromic complexity of trees. For standard strings, i.e., for paths, the palindromic complexity is $$n+1$$.

Tight Bound for the Number of Distinct Palindromes in a Tree