The number of size-n recursive trees is given by Hence the exponential generating function T(z) 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.
denotes the node labeled by 1, × the Cartesian product and
By translation of the formal description one obtains the differential equation for T(z) with T(0) = 0.
Recursive trees can be generated using a simple stochastic process.
Such random recursive trees are used as simple models for epidemics.