Recursive tree


In graph theory, a recursive tree is a non-planar labeled rooted tree. A size-n recursive tree is labeled by distinct 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 node are not ordered. E.g. the following two size-three recursive trees are the same.

 1 1
 / \ = / \
 / \ / \
 2 3 3 2

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

Properties

The number of size-n recursive trees is given by
Hence the exponential generating function T of the sequence Tn is given by
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.
where denotes the node labeled by 1, × the Cartesian product and the partition product for labeled objects.
By translation of the formal description one obtains the differential equation for T
with T = 0.

Bijections

There are bijective correspondences between recursive trees of size n and permutations of size n − 1.

Applications

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