prepositions - "He committed a crime killing/by killing a bird"

Which wording with kill should be used here:

He committed a crime (killing/by killing) a bird.

My guess is the answer would be by killing. But I am really confused with the answer killing.

My questions relating to this sentence are:

1. 
   

   Which one is correct and why? ^{(or are they both correct?)}

   
   


2. 
   

   If one of them is incorrect, why is it incorrect? ^{(or, if both of them are, why?)}

   
   


3. 
   

   If they are both correct, what is the difference in meaning?

   
   



4. 
   

   If they are both correct, what is the difference in their grammar?