I don't think this article is referring to physical organisms known as trees (as your article does). It's talking about abstract tree structures, in particular:
1. There is a coherent parent-child relationship
2. Children have only one parent
3. Children have no lateral relationships with other children
4. Relationships are mediated through parents, not e.g. grandparent-child directly.
5. We can coherently differentiate between different children
etc.
All of these are assumptions that greatly simplify analysis and therefore useful in any situations, but almost always fail in some way on closer inspection.
More generally we go to the saying "all models are wrong, some are useful."
I'm 95% sure that the author (who is on HN[1]) is at least referring to the article "There's No Such Thing As A Tree" in his line "There is no such thing as a tree." - regardless of the fact that the article as a whole isn't about trees made of wood.
It's possible that he's additionally making a double entendre about abstract tree structures.
1. There is a coherent parent-child relationship
2. Children have only one parent
3. Children have no lateral relationships with other children
4. Relationships are mediated through parents, not e.g. grandparent-child directly.
5. We can coherently differentiate between different children
etc.
All of these are assumptions that greatly simplify analysis and therefore useful in any situations, but almost always fail in some way on closer inspection.
More generally we go to the saying "all models are wrong, some are useful."