A city is not a tree – Christopher Alexander
The tree of my title is not a green tree with leaves. It is the name of an abstract structure. I shall contrast it with another, more complex abstract structure called a semi lattice. In order to relate these abstract structures to the nature of the city, I must first make a simple distinction.
I want to call those cities which have arisen more or less spontaneously over many, many years natural cities. And I shall call those cities and parts of cities which have been deliberately created by designers and planners artificial cities. Siena, Liverpool, Kyoto, Manhattan are examples of natural cities. Levittown,Chandigarh and the British New Towns are examples of artificial cities.
It is more and more widely recognized today that there is some essential ingredient missing from artificial cities. When compared with ancient cities that have acquired the patina of life, our modern attempts to create cities artificially are, from a human point of view, entirely unsuccessful.
Both the tree and the semi lattice are ways of thinking about how a large collection of many small systems goes to make up a large and complex system.More generally, they are both names for structures of sets.
In order to define such structures, let me first define the concept of a set. A set is a collection of elements which for some reason we think of as belonging together. Since, as designers, we are concerned with the physical living city and its physical backbone, we must naturally restrict ourselves to considering sets which are collections of material elements such as people, blades of grass, cars, molecules, houses, gardens, water pipes, the water molecules in them etc.
When the elements of a set belong together because they co-operate or work together somehow, we call the set of elements a system.
For example, in Berkeley at the corner of Hearst and Euclid, there is a drugstore, and outside the drugstore a traffic light. In the entrance to the drugstore there is a news rack where the day’s papers are displayed. When the light is red, people who are waiting to cross the street stand idly by the light;and since they have nothing to do, they look at the papers displayed on the news rack which they can see from where they stand. Some of them just read the headlines, others actually buy a paper while they wait.
This effect makes the news rack and the traffic light interactive; the news rack,the newspapers on it, the money going from people’s pockets to the dime slot,the people who stop at the light and read papers, the traffic light, the electric impulses which make the lights change, and the sidewalk which the people stand on form a system – they all work together.
From the designer’s point of view, the physically unchanging part of this system is of special interest. The news rack, the traffic light and the sidewalk between them, related as they are, form the fixed part of the system. It is the unchanging receptacle in which the changing parts of the system – people,newspapers, money and electrical impulses – can work together. I define this fixed part as a unit of the city. It derives its coherence as a unit both from the forces which hold its own elements together and from the dynamic coherence of the larger living system which includes it as a fixed invariant part.
Of the many, many fixed concrete subsets of the city which are the receptacles for its systems and can therefore be thought of as significant physical units, we usually single out a few for special consideration. In fact, I claim that whatever picture of the city someone has is defined precisely by the subsets he sees as units.
Now, a collection of subsets which goes to make up such a picture is not merely an amorphous collection. Automatically, merely because relationships are established among the subsets once the subsets are chosen, the collection has a definite structure.
To understand this structure, let us think abstractly for a moment, using numbers as symbols. Instead of talking about the real sets of millions of real particles which occur in the city, let us consider a simpler structure made of just half a dozen elements. Label these elements 1,2,3,4,5,6. Not including the full set [1,2,3,4,5,6], the empty set [-], and the one-element sets,,,,, , there are 56 different subsets we can pick from six elements.
Suppose we now pick out certain of these 56 sets (just as we pick out certain sets and call them units when we form our picture of the city). Let us say, for example, that we pick the following subsets: , , , , ,, .
What are the possible relationships among these sets? Some sets will be entirely part of larger sets, as  is part of  and . Some of the sets will overlap, like  and . Some of the sets will be disjoint – that is,contain no elements in common like  and .
We can see these relationships displayed in two ways. In diagram A each set chosen to be a unit has a line drawn round it. In diagram B the chosen sets are arranged in order of ascending magnitude, so that whenever one set contains another (as  contains ,there is a vertical path leading from one to the other. For the sake of clarity and visual economy, it is usual to draw lines only between sets which have no further sets and lines between them; thus the line between  and  and the line between  and  make it unnecessary to draw a line between and .
As we see from these two representations, the choice of subsets alone endows the collection of subsets as a whole with an overall structure. This is the structure which we are concerned with here. When the structure meets certain conditions it is called a semi lattice. When it meets other more restrictive conditions, it is called a tree.
The semi lattice axiom goes like this: A collection of sets forms a semi lattice if and only if, when two overlapping sets belong to the collection, the set of elements common to both also belongs to the collection.
The structure illustrated in diagrams A and B is a semi lattice. It satisfies the axiom since, for instance,  and  both belong to the collection and their common part, , also belongs to it. (As far as the city is concerned, this axiom states merely that wherever two units overlap, the area of overlap is itself a recognizable entity and hence a unit also. In the case of the drugstore example, one unit consists of news rack, sidewalk and traffic light. Another unit consists of the drugstore itself, with its entry and the news rack. The two units overlap in the news rack. Clearly this area of overlap is itself a recognizable unit and so satisfies the axiom above which defines the characteristics of a semi lattice.) The tree axiom states: A collection of sets forms a tree if and only if, for any two sets that belong to the collection either one is wholly contained in the other, or else they are wholly disjoint.