In 1988, Robert O’Hara coined the now ubiquitous phrase “tree thinking” to highlight the importance of cladistics for proper evolutionary reasoning. This accessible phrase has been taken up widely in the professional, popular, and educational literatures, and it has played an important role in helping spread phylogenetic thinking far beyond the disciplinary borders of systematics. However, the undeniable benefits of the spread of tree thinking have become marred by being widely linked to several misconceptions that were present in O’Hara’s original writings. O’Hara incorrectly considered clades to be the central subjects of evolutionary narratives. By failing to appreciate that clades contain independently evolving lineages, O’Hara has promoted the misleading view that evolution is irreducibly branched. In this paper, I show how an exclusive focus on the branching realm of taxa has created a cladistic blindfold that has caused a form of lineage blindness that has spread widely through the literature dedicated to the teaching of tree thinking. Its symptoms include the rejection of phenomena and concepts that are fundamental to the realm of evolving lineages, including linear evolutionary imagery and narratives, the concepts of anagenetic evolution and missing links, our evolutionary descent from monkeys and apes, and the promotion of the nonsensical concept of collateral ancestors. To avoid simplistic tree thinking, it is crucial to recognize that the realms of taxa and lineages have distinctive features that require different kinds of thinking. I close by suggesting that teaching can be improved by linking tree thinking explicitly to lineage thinking.