When we think of a city, we might picture it as a tidy arrangement of separate parts: residential districts here, shopping areas there, parks in another place, schools set apart from daily circulation. The image feels intuitive. It implies order, clarity, and an apparent kind of efficiency. 

Or we might picture something quite different: streets where shops sit beneath apartments, workshops spill into alleys, and public spaces blend into daily movement rather than being set aside. In many parts of the world, this dense mix of fragments feels just as natural as strict separation, perhaps even inevitable, depending on the city that shaped our habits and expectations.

So what kind of structure truly captures how urban spaces work? Does it follow clear separations, or does it emerge from overlaps?

In 1965, architect Christopher Alexander published a short essay with a striking title: A city is not a tree [1]. The essay did not simply criticize planning trends of his time; it questioned the very way we imagine the structure of a city. What kind of form do we believe a city has? And what kind of form does it actually need to support everyday life?

To explore these questions, Alexander turned, perhaps surprisingly, to mathematics (and spoiler: the trees in his title are not oaks, firs, or willows, but mathematical structures!). As we noted in the previous article, besides being analyzed through its forms and geometries, a city can also be studied by looking at how its parts relate to one another. When the focus is on relationships rather than only on shapes, a powerful tool becomes available: the theory of graphs. Graphs represent a network through points and the connections between them, allowing us to describe the structure of the city in terms of how its elements interact.

Yet not all graphs describe the same kind of relationships. Some organize elements into distinct categories, while others allow those categories to overlap. These differences matter, because they lead to different interpretations of how urban environments are structured.

Among the many possible graphs, two stand out as particularly meaningful for talking about cities: trees and semi-lattices. Both provide ways of organizing relationships, yet they point to contrasting ways in which urban life takes form.

Trees: when a city is divided into single purposes

In graph theory, a tree is a graph where connections branch outward without looping back, and each point belongs to one and only one branch. It is a hierarchy of clear lines: parents, children, subcategories, subdivisions. Nothing overlaps; every element has its single place within the diagram.

Throughout the twentieth century, many urban plans were conceived in this way. Under the banner of efficiency and clarity, planners proposed cities made of large residential sectors, distant business districts, isolated green belts, and specialized roads. The CIAM (Congrès Internationaux d'Architecture Moderne) vision of the “Functional City” promoted four distinct zones: dwelling, work, leisure, circulation [2]. Even new capitals such as Brasília were designed as sequences of designated areas, each performing its own task, linked but not interwoven.

A similar logic shaped numerous developments in the United States, where suburban subdivisions, shopping centers, office parks, and school campuses were planned as discrete clusters rather than interwoven fabrics. These environments are often described as artificial cities: not because they are less real, but because their structure does not emerge from gradual adaptation. Instead, it is imposed from above, as a system of clearly separated units.

Semi-Lattices: When Urban Life Overlaps

A semi-lattice is also a graph, but it allows relationships that a tree forbids. In a semi-lattice, elements can belong to more than one set at once, and the intersections between sets are part of the structure, not an anomaly.

Intersections as a form of order

Trees and semi-lattices are both logical ways of organizing a city, but they lead to very different interpretations of what a city is. A tree separates; a semi-lattice intertwines. Seen this way, overlap is not disorder. It expresses a different kind of structure: one based on relationships that accumulate rather than on functions that are kept apart. A semi-lattice does not sort elements into separate branches; it lets them participate in several networks at once, and it is from these overlaps that urban life gains depth.

And what is unusual is that the mathematics is not produced in order to prove a theorem; it is borrowed as a language to distinguish two forms that may look similar in practice but behave differently in structure. Tree and semi-lattice were not invented for cities, yet they reveal differences that drawings or definitions alone would fail to make explicit.

If we follow Alexander’s reasoning, then understanding a city means recognizing how different activities interact, not only how they are separated. It means paying attention to places that overlap, rather than to those assigned to a single use. It suggests that the life of a city might depend less on the clarity of its branches and more on the density of its intersections.

But if that is true, then a new question emerges: what kind of systems behave this way? What mathematics can describe structures that are neither fully ordered nor fully chaotic, but shaped by ongoing interactions between many small decisions?

To answer this, we must move beyond trees and semi-lattices into the territory of complex systems, where patterns emerge from the bottom up, and where the statistics of everyday life leave their trace in the shape of the city.

That will be the subject of the next article.

[1] Alexander, Christopher. A city is not a tree. Architectural forum, 1965.

[2] Mumford, Eric. The CIAM discourse on urbanism, 1928–1960. MIT Press, 2000 

Illustrations by Laura Govoni - visual interpretations of mathematics.