If the player's melee weapon is any form of axe and they chop down a tree, a long-stick is automatically placed in the player's melee back/sling spot because their melee slot weapon is currently in their hands to chop down the tree.

If the player uses a hotkey to swap or de-equip their axe, it is simply dropped on the ground.

I believe the correct behavior should be that when chopping down a tree, nothing is equipped.