Regular expression for a binary string containing even number of 0's To get the regular expression I made a finite automata as the following (not sure if you can directly write regular expression without it): 



The regular expression for the above according to me should be $(1+01^*0)^*$ but elsewhere I have seen it can be $(1^*01^*01^*)^*$. Why is it different?
(1∗01∗01∗)∗ is actually wrong because it doesn't handle a string consisting of all 1's