Do you have time? Are you relatively safe from zombies for a while? Are you ok with this taking longer?
Ok... you don't need an axe. Any weapon can take down a tree, eventually. Axes are just much much faster.
A hammer will get the job done in about 4 * the time; it goes downhill from there... nailbat is better than golf club. Golf club is better than kitchen knife.
Don't ask about the fork. Or do, what do I care. It's my signature after all. True story.
160 hits with a fork will give you 2 logs from a tree... and take forever.
And the tines of the fork; bent all to hell.
Don't believe me? Start a new sandbox game without zombies; spend an in-game hour whaling on a tree with a fork... laugh as it goes down.
I was doing some tests with "strength xp" and trying to remove the negative strength perks to see what that would take.
It takes beating a lot of trees with a fork. For a long long time.