Palindromes in starlike trees.

In this note, we obtain an upper bound on the maximum number of distinct non-empty palindromes in starlike trees. This bound implies, in particular, that there are at most 4n distinct non-empty palindromes in a starlike tree with three branches each of length n. For such starlike trees labelled with a binary alphabet, we sharpen the upper bound to 4n-1 and conjecture that the actual maximum is 4n-2. It is intriguing that this simple conjecture seems difficult to prove, in contrast to the proof of the bound.