I found a potential softlock - if you use Land as a bridge to the next island instead of Bridge, it is possible for a tree to spawn in the one-tile wide strip of land. And if it happens when you don't have a pickaxe on yourself... you're screwed.
Yeah it's unlikely but it is, theoretically possible. I think biggest chance for this to happen is upgrading your pickaxe with full inventory - when you upgrade your pick it vanishes from your inventory. And if you have full inventory when upgrade is over, it would just fall on the ground.