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?

k, this isnt mine, but my friends: (7 strokes)

2 heads (nothing)

1 tail (adds 2 tails)

1 tail (2 more tails)

2 tails (adds a head)

2 tails (another head)

2 heads (nothing)

poke him in the belly, making him bend in half, and cut off the last  head and last tail in the same stroke. cha-ching!