Christopher Alexander: A City is not a Tree part 1
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'A City is not a Tree'
by Christopher Alexander
A CITY IS NOT A TREE
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 semilattice. 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 semilattice 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 newsrack 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 newsrack 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 newsrack and the traffic light interactive; the newsrack, 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 newsrack, 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 ,,,C4],, , 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 .
Diagrams A & B redrawn by Nikos Salingaros
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 semilattice. When it meets other more restrictive conditions, it is called a tree.
The semilattice axiom goes like this: A collection of sets forms a semilattice 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 semilattice. 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 newsrack, sidewalk and traffic light. Another unit consists of the drugstore itself, with its entry and the newsrack. The two units overlap in the newsrack. Clearly this area of overlap is itself a recognizable unit and so satisfies the axiom above which defines the characteristics of a semilattice.) 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.
The structure illustrated in diagrams C and D is a tree. Since this axiom excludes the possibility of overlapping sets, there is no way in which the semilattice axiom can be violated, so that every tree is a trivially simple semilattice.
Diagrams A & B redrawn by Nikos Salingaros
However, in this chapter we are not so much concerned with the fact that a tree happens to be a semilattice, but with the difference between trees and those more general semilattices which are not trees because they do contain overlapping units. We are concerned with the difference between structures in which no overlap occurs, and those structures in which overlap does occur.
It is not merely the overlap which makes the distinction between the two important. Still more important is the fact that the semilattice is potentially a much more complex and subtle structure than a tree. We may see just how much more complex a semilattice can be than a tree in the following fact: a tree based on 20 elements can contain at most 19 further subsets of the 20, while a semilattice based on the same 20 elements can contain more than 1,000,000 different subsets.
This enormously greater variety is an index of the great structural complexity a semilattice can have when compared with the structural simplicity of a tree. It is this lack of structural complexity, characteristic of trees, which is crippling our conceptions of the city.
To demonstrate, let us look at some modern conceptions of the city, each of which I shall show to be essentially a tree.
Figure 1. Columbia, Maryland, Community Research and Development, Inc.: Neighbourhoods,in clusters of five, form 'villages'. Transportation joins the villages into a new town. The organization is a tree.
Figure 2. Greenbelt, Maryland, Clarence Stein: This 'garden city' has been broken down into superblocks. Each superblock contains schools, parks and a number of subsidiary groups of houses built around parking lots. The organization is a tree.
Figure 3. Greater London plan (1943), Abercrombie and Forshaw: The drawing depicts the structure conceived by Abercrombie for London. It is made of a large number of communities, each sharply separated from all adjacent communities. Abercrombie writes, 'The proposal is to emphasize the identity of the existing communities, to increase their degree of segregation, and where necessary to recognize them as separate and definite entities.' And again, 'The communities themselves consist of a series of sub-units, generally with their own shops and schools, corresponding to the neighbourhood units.' The city is conceived as a tree with two principal levels. The communities are the larger units of the structure; the smaller sub-units are neighbourhoods. There are no overlapping units. The structure is a tree.
Figure 4. Mesa City, Paolo Soleri: The organic shapes of Mesa City lead us, at a careless glance, to believe that it is a richer structure than our more obviously rigid examples. But when we look at it in detail we find precisely the same principle of organization. Take, particularly, the university centre. Here we find the centre of the city divided into a university and a residential quarter, which is itself divided into a number of villages (actually apartment towers) for 4000 inhabitants, each again subdivided further and surrounded by groups of still smaller dwelling units.
Figure 5. Tokyo plan, Kenzo Tange: This is a beautiful example. The plan consists of a series of loops stretched across Tokyo Bay. There are four major loops, each of which contains three medium loops. In the second major loop, one medium loop is the railway station and another is the port. Otherwise, each medium loop contains three minor loops which are residential neighbourhoods, except in the third major loop where one contains government offices and another industrial offices.
Figure 6. Chandigarh (1951), Le Corbusier: The whole city is served by a commercial centre in the middle, linked to the administrative centre at the head. Two subsidiary elongated commercial cores are strung out along the maior arterial roads, running north-south. Subsidiary to these are further administrative, community and commercial centres, one for each of the city's 20 sectors.
Figure 7. Brasilia, Lucio Costa: The entire form pivots about the central axis, and each of the two halves is served by a single main artery. This main artery is in turn fed by subsidiary arteries parallel to it. Finally, these are fed by the roads which surround the superbiocks themselves. The structure is a tree.
Figure 8. Communitas, Percival and Paul Goodman: Communitas is explicitly organized as a tree: it is first divided into four concentric major zones, the innermost being a commercial centre, the next a university, the third residential and medical, and the fourth open country. Each of these is further subdivided: the commercial centre is represented as a great cylindrical skyscraper, containing five layers: airport, administration, light manufacture, shopping and amusement; and, at the bottom, railroads, buses and mechanical services. The university is divided into eight sectors comprising natural history, zoos and aquariums, planetarium, science laboratories, plastic arts, music and drama. The third concentric ring is divided into neighbourhoods of 4000 people each, not consisting of individual houses, but of apartment blocks, each of these containing individual dwelling units. Finally, the open country is divided into three segments: forest preserves, agriculture and vacation lands. The overall organization is a tree
Figure 9.The most beautiful example of all I have kept until last, because it symbolizes the problem perfectly. It appears in Hilberseimer's book The Nature of Cities. He describes the fact that certain Roman towns had their origin as military camps, and then shows a picture of a modern military encampment as a kind of archetypal form for the city. It is not possible to have a structure which is a clearer tree. The symbol is apt, for, of course, the organization of the army was designed precisely in order to create discipline and rigidity. The photograph on the [left] is Hilberseimer's own scheme for the commercial area of a city based on the army camp archetype.
Each of these structures, then, is a tree. Each unit in each tree that I have described, moreover, is the fixed, unchanging residue of some system in the living city (just as a house is the residue of the interactions between the members of a family, their emotions and their belongings; and a freeway is the residue of movement and commercial exchange).
However, in every city there are thousands, even millions, of times as many more systems at work whose physical residue does not appear as a unit in these tree structures. In the worst cases, the units which do appear fail to correspond to any living reality; and the real systems, whose existence actually makes the city live, have been provided with no physical receptacle.
Christopher ALEXANDER: A city is not a tree
© Christopher Alexander