Recursive tree

In graph theory, a recursive tree (i.e., unordered tree) is a labeled, rooted tree. A size-$n$ recursive tree's vertices are labeled by distinct positive integers $1, 2, …, n$, where the labels are strictly increasing starting at the root labeled 1. Recursive trees are non-planar, which means that the children of a particular vertex are not ordered; for example, the following two size-3 recursive trees are equivalent: $3/1\2 = 2/1\3$.

Recursive trees also appear in literature under the name Increasing Cayley trees.

Properties
The number of size-n recursive trees is given by


 * $$ T_n= (n-1)!. \,$$

Hence the exponential generating function T(z) of the sequence Tn is given by


 * $$ T(z)= \sum_{n\ge 1} T_n \frac{z^n}{n!}=\log\left(\frac{1}{1-z}\right).$$

Combinatorically, a recursive tree can be interpreted as a root followed by an unordered sequence of recursive trees. Let F denote the family of recursive trees. Then


 * $$ F= \circ + \frac{1}{1!}\cdot \circ \times F

+ \frac{1}{2!}\cdot \circ \times F* F + \frac{1}{3!}\cdot \circ \times F* F* F * \cdots = \circ\times\exp(F),$$

where $$\circ$$ denotes the node labeled by 1, &times; the Cartesian product and $$*$$ the partition product for labeled objects.

By translation of the formal description one obtains the differential equation for T(z)


 * $$ T'(z)= \exp(T(z)),$$

with T(0) = 0.

Bijections
There are bijective correspondences between recursive trees of size n and permutations of size n &minus; 1.

Applications
Recursive trees can be generated using a simple stochastic process. Such random recursive trees are used as simple models for epidemics.