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Concept

Labelled tree

A tree — a connected network with no cycles — whose vertices carry distinct names, so two trees are the same only if the same pairs of names are joined. There are n^(n−2) of them on n names, Cayley's formula, and the labels are what make that count finite and exact.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

All 16 trees on 4 labelled points. Every tree on 4 labelled points, drawn one by one. There are 16 of them, which is 4 to the power 2.

Sixteen trees on four points

How many ways are there to connect n labelled points into a single tree? The answer is n to the power n minus two, which is a strange enough formula to demand an explanation — and the explanation is a code that turns every tree into a short list of numbers, and every short list of numbers back into a tree.

discrete · Labelled trees Orbit times stabiliser is 8, on every row. A table of 5 things the 8 symmetries of a 4-gon can move. Each row draws every position the thing can be carried to and lists the motions that leave it where it is; the two counts multiply to 8 on every row.

Twenty-four ways to set a cube down

Count the rotations of a cube from its corners and the answer is eight times three. Count from its edges and it is twelve times two; from its faces, six times four. Three different pictures give one number because each count is the same theorem — the places a thing can go, times the motions that leave it where it is — and the same theorem splits Cayley's sixteen trees into twelve and four and proves that a group of eight has a centre.

algebra · Symmetry groups Pollak's circle: one rotation in every n + 1 parks on the line. Several circles of numbered spots, each showing where cars park when every preference in a list is rotated by a fixed amount, with the one rotation that leaves the last spot empty highlighted.

Cars that park, and trees that grow

Three cars arrive at a one-way street with three spaces; each has a favourite space, drives to it, and takes the first free one from there on. Of the 27 lists of favourites, exactly 16 let every car park — the same 16 as the labelled trees on four points. The reason is a circular street with one extra space, on which every list parks and exactly one rotation of it leaves the extra space empty.

discrete · Labelled trees A random labelled tree on 60 points. A tree drawn in horizontal layers by distance from a root point, with the leaves coloured differently from the internal points.

A random tree is one part in e leaves

Choose a labelled tree on n points uniformly at random. A point is a leaf exactly when its label never appears in the tree's Prüfer code, so the share of leaves is (1 − 1/n)^(n − 2) — half the points for a tree on four, 36.8% for a large one, the reciprocal of e. The whole degree distribution follows the same way: one plus a Poisson count with mean one.

discrete · Labelled trees A function on nine points, and the tree with two marks it becomes. Left: a function on 9 points as arrows, with cycles through 2, 5, 7 and trees hanging into them. Right: the tree Joyal's rule makes from it, a path from head 7 to tail 2 with the same trees hanging below.

Every function is a tree with two marks

There are n to the n functions from n points to themselves, and n to the n − 2 trees on those points. André Joyal noticed in 1981 that the missing factor of n² is a choice of two points — a head and a tail — and that a function, read the right way, simply is a tree with a head and a tail. The reading turns Cayley's formula into one line and hands over a fact about random trees from the birthday problem.

discrete · Labelled trees The 11 trees on 7 points, and how many labellings each has. All trees on 7 points with their numbers of symmetries and of distinct labellings; the labellings add to 16807.

Almost every tree can be turned over

Cayley's n^(n−2) counts trees with labels on their points. Take the labels off and the count has no formula, because a symmetric shape absorbs labellings: the star on seven points can be labelled only seven ways, the one asymmetric shape 5,040. The bookkeeping that reconciles the two counts says something unexpected — almost every tree, labelled or not, can be turned over onto itself, where almost every graph cannot.

discrete · Labelled trees

Named alongside it

The objects these essays reach for when they reach for this one.

Exhaustive searchBijectionCounting argumentAutomorphismCayleys formulaCounting two waysExpectationGroup actionLeafPermutationPrufer codeSpanning tree

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