What does it mean that hierarchies are not trees they are lattices?

Real knowledge often has items that belong to multiple parent categories. When you force every concept into a single branch of a tree, you destroy information. Lattice structures — where a node can have multiple parents — preserve the multidimensional nature of knowledge. The tree is a special case. The lattice is the general case.

Example: A physician diagnosing a patient with liver cancer opens SNOMED CT, the clinical terminology system used in healthcare worldwide. She navigates to "Neoplasm of liver." In SNOMED CT, this concept does not sit in one branch. It is simultaneously a subtype of "Disorder of liver" and a subtype of "Neoplasm." She can reach it by navigating down from organ systems or down from disease types. This is not a quirk of SNOMED CT — it is a design principle called polyhierarchy. The concept genuinely belongs in both places because it genuinely is both things: a liver disorder and a neoplasm. Forcing it into one branch would destroy clinical information. A physician searching by organ would miss it if it were filed only under neoplasm type. A oncologist searching by disease category would miss it if it were filed only under liver disorders. The polyhierarchy preserves both access paths because both are true.

Try this: Choose a domain you organize — your notes, your project files, your reading list, your skill inventory. Pick five items and ask: does each item have exactly one parent, or does it genuinely belong in multiple categories? For each item with multiple natural parents, write down all the parents it belongs to. Then draw the structure — nodes for the items, directed edges pointing upward to each parent. Notice how the result is not a tree. It is a directed acyclic graph. Now ask: what information would you lose if you were forced to assign each item to exactly one parent? What retrieval paths would break? Write a paragraph describing what the lattice structure preserves that the tree structure destroys.