Can you use a tree diagram to calculate the probability...?

Yes you most certainly can!


  1
   2
1   3
    4
   5
  6


  1
   2
2   3
    4
   5
  6


  1
   2
3   3
    4
   5
  6


  1
   2
4   3
    4
   5
  6


  1
   2
5   3
    4
   5
  6


  1
   2
6   3
    4
   5
  6

It's a little hard to show it on a computer, but now trace each branch and look for even numbers!
You could already tell the answer is ½, through this logic:
If you have an odd number on the first dice, you would need another odd number on the second dice for the sum to be even.  
IF you have an even number on the first dice, you'd need an even on the second, also.  Since there are an even number of numbers (1,2?5,6), it will be a fifty-fifty percent chance of getting even (or odd).

Hope this helps:)

ogyno is correct.  You can use a tree diagram to map out every possibility of the rolls.