Hi everyone!
As you may know, it is possible to build a suffix automaton for a set of strings as well as suffix tree consisting of all suffixes of strings from the set. Question is as follows: for each string Sk consider set Vk of vertices/states of tree/automaton that corresponds to the substrings of Sk. Is it true that 
? Can you prove it or make a counter-example?
What prevents |Vk| from being always Θ(|Sk|) for suffix trees?
We can take Sk, k > 1 as set of substrings of S1. Then |V1| will be Θ(|S1|2).
Set of prefixes should have the same effect. Every |Vk| is Ω(|Sk|2) then, right?
Seems to be correct, cool! Then for total length
this sum will be 
and 
. Can you improve the boundaries now?