We consider the continuous-time version of the random digital search tree, and construct a coupling with a border aggregation model as studied in Thacker and Volkov (Ann. Appl. Probab. 28 (2018) 1604–1633), showing a relation between the height of the tree and the time required for aggregation. This relation carries over to the corresponding discrete-time models. As a consequence we find a very precise asymptotic result for the time to aggregation, using recent results by Drmota et al. (Random Structures Algorithms 58 (2021) 430–467) for the digital search tree.
Svante Janson. Debleena Thacker. "Continuous-time digital search tree and a border aggregation model." Bernoulli 28 (4) 2563 - 2577, November 2022. https://doi.org/10.3150/21-BEJ1429