An article by Christopher Alexander, published in two installments (April and May) in Architectural Forum in 1965. The “tree” in the title means a tree structure. The author contrasts it with another mathematical structure, the semilattice.

Introduction

Before discussing structure further, he begins by distinguishing two kinds of cities.

Natural city: A city that arose naturally
Artificial city: A city designed in modern times through urban planning and the like

The author feels that artificial cities lack some essential ingredient present in natural cities. Unlike ancient natural cities, modern artificial cities lack the patina of life. (“Patina of life” probably connects to what he later calls the Quality without a name. —ak)

Many contemporary architectural designers try to solve this by imitating the appearance of natural cities. But it is important to find a more abstract principle of order (The nature of order?) rather than appearances. The essential difference is that natural cities have semilattice structures, whereas artificial cities have tree structures.

Trees and semilattices

When different elements work together in some way to perform a function, we call their collection a system. For example, a shop stands at an intersection, with a newsstand at its entrance. Pedestrians waiting for the crossing signal have nothing else to do, so they read or buy newspapers. This is a system comprising newspapers, the stand, pedestrians, traffic lights, the crosswalk, coins, electricity, and so on.

For urban designers, the unchanging, static elements are particularly important, because they form the vessels for the dynamic events unfolding in the system. C.A. calls a static part of a city a unit. The cohesion that makes a unit a unit arises from the forces binding its elements and the dynamic cohesion of the larger system containing it.

C.A. contrasts two recursive ways in which elements form units: trees and semilattices. For example, the following shows the difference between two structures taking certain subsets (sub-elements) of a system consisting of six elements, {1,2,3,4,5,6}. The first structure is a semilattice, the second a tree.

In a semilattice, whenever two units overlap, their overlapping region is itself always another unit in the structure. For example, if there is a unit {shop, newsstand} and another {newsstand, traffic light, crosswalk}, their intersection {newsstand} must also be a unit.
In a tree, an element belonging to one structure cannot belong to another element. (By definition, all trees are semilattices, but this article contrasts trees with semilattices that are not trees.)

A tree with 20 elements can have at most 19 subsets, whereas a semilattice can have more than a million. Artificial cities are inherently simple because they have tree structures, while natural cities have semilattice structures. This is the greatest distinction between them.

Whether a city looks organic is irrelevant. What matters is whether the part-whole relationships among its units form a tree or a semilattice. In a tree, an element belongs to only one unit and cannot connect to other units in any other way.

The idea of a clear boundary separating a university campus from the rest of a town is unnatural. In Cambridge, where city and university grew naturally together, physical units overlap in a semilattice structure.

Cities designed as trees fragment people’s lives

Trees do not adequately describe the actual structure of natural cities. Designers nevertheless design cities as trees because the human mind struggles with the complexity of semilattices. Every time part of a natural city is replaced by an artificial city, the city becomes more dissociated. The first sign of destruction in a structured object is extreme fragmentation and dissociation of its internal elements. Social dissociation is anarchy; individual dissociation (fragmentation) is a characteristic of schizophrenia.

Trees are useful for holding complex ideas in the human mind, but a city is not a tree. A city is a vessel for life. A city designed as a tree is like a bowl full of razor blades: whatever you put into it is bound to be cut to pieces. Designing a city as a tree fragments the lives of the people who live there. #

Personal notes

This applies to many things beyond cities. Many things are not trees.