Rafaello's tree grows according to the following rule. After a branch has been growing for two weeks, it produces a new branch every week, while the original branch continues to grow. The tree has five branches after five weeks, as shown. How many branches, including the main branch, will the tree have at the end of eight weeks?
(A) 21
(B) 40
(C) 19
(D) 13
(E) 34
Every week, a branch that is at least two weeks old produces a new branch. Therefore, after a given week, the total number of branches is equal to the number of branches at the beginning of the week plus the number of branches that were not new during the previous week (ie. the number of "old" branches). So we make a chart:

| Week \# | Branches at beginning | Old branches at beginning | Branches at end |
| :---: | :---: | :---: | :---: |
| 1 | 0 | 0 | 1 |
| 2 | 1 | 0 | 1 |
| 3 | 1 | 1 | 2 |
| 4 | 2 | 1 | 3 |
| 5 | 3 | 2 | 5 |
| 6 | 5 | 3 | 8 |
| 7 | 8 | 5 | 13 |
| 8 | 13 | 8 | 21 |

Therefore, there are 21 branches at the end of the eighth week.

Answer: (A)