Sadly the way this works is there seems to be 2 tiers of rare items. The soft sap is the green tier I for the young trees or trees from 1-50. You generally start seeing these at lumber level 40+.
So as for the petrified wood this is a tier II green item as denoted by the “II”. This would be the green item from the 50-100 level stuff. It does drop from both young and mature trees but also you won’t see it drop UNTIL you are within 10ish levels of the next tier aka lumber level 90.
Once you are 90+ you will see these start showing up. Until then don’t count on it. Not sure if it’s a bug or I was already above 90 in alpha but I do know you get them at some point. I also spent a good bit of time leveling lumber to try and get these and none so far with both lumber food and gathering potion at level 67.
Just food for thought to keep you from going crazy.