We show that an algorithmic construction of sequences of recursive trees leads to a direct proof of the convergence of random recursive trees in an associated Doob-Martin compactification; it also gives a representation of the limit in terms of the input sequence of the algorithm. We further show that this approach can be used to obtain strong limit theorems for various tree functionals, such as path length or the Wiener index.
"Random recursive trees: a boundary theory approach." Electron. J. Probab. 20 1 - 22, 2015. https://doi.org/10.1214/EJP.v20-3832