He has a magic sword that can, in one stroke, chop off either one head, two heads, one tail, or two tails.

This dragon is of a type related to the hydra; if one head is chopped off, a new head grows. In place of one tail, two new tails grow; in place of two tails, one new head grows; if two heads are chopped off, nothing grows.

What is the smallest number of strokes required to chop off all the dragon's heads and tails, thus killing it?

i think its 9 chopping

why? because the last thing to do is to chop 2 heads...its the only way to eliminate both heads and tails. so the heads must be even before we can do it. chop 1 head is useless. the first step is to chop 1 tail or 2 tails. to eliminate tails we must make the tail even ( 2, 4, 6, ...) in condition amount of tail divided by 2 is odd because we just need odd amount to make amount of heads even.  so we need to chop one tail 3 times, two tail 3 times, and two heads 3 times.