Count the number of distinct palindromic substrings in a string

First distinguish counting palindromic occurrences from counting distinct palindrome strings. For small constraints, expand around every odd and even center and insert each discovered substring into a hash set; it is easy to explain but can cost O(n cubed) time if substring creation/comparison is not constant. For a scalable answer, use a palindromic tree (Eertree): nodes represent distinct palindromes, suffix links find the longest palindromic suffix after appending each character, and each new node represents one new distinct palindrome. That yields O(n) nodes and near O(n) processing for a fixed alphabet. Mention Manacher alone counts radii but does not deduplicate strings. Test repeated characters, an empty string, and even palindromes. Pick the implementation only after confirming constraints and language support.