Conifers aren't trees by this definition. It seems to completely ignore gymnosperms and even misclassified a couple as dicots like sequoias and junipers.
We need to stop looking for a scientifically coherent category for a tree and ,like fish, embrace the true, intuitive, childlike definition of it as just a form, a trunk with leaves at the top.
If I remember correctly, wood consists mainly of cellulose, lignin, and hemi-cellulose. I don't know about bamboo, but I guess it's some kind of woody material.