I have 2 questions in mind, just out of curiousity :). They both consider tries (not compressed). Also note — by a trie's size I'm not considering its root (even though, it does not matter for the purpose of this problem).
1) For some 2 natural numbers N and K (suppose K ≤ N), what is the maximal size of a trie that can be achieved if you take some string S of length N, with the size of its language (number of distinct characters it can contain) being K, and insert into the trie all its suffixes (so, a suffix trie)?
For instance, if N = 2, K = 2, we can form the string "ab", which makes the trie of size 3. On the other hand, if N = 10, K = 1, the only string we can form is "aaaaaaaaaa" which makes the trie size 10.
2) How do you form such a string with maximal trie size? An algorithm or an idea, both are acceptable.