The Sackin index and depth of leaves in generalized Schröder trees

Schröder trees are biological models of evolution, with internal nodes having two or three children. We generalize the model to grow from an arbitrary stochastic process of independent nonnegative integers (not necessarily identically distributed). We call such a process the building sequence. We study the depth of leaves and the Sackin index for some specific building sequences, such as constant additions, Bernoulli, and Poisson-like models. We include an example that shows that the methods can be extended to exchangeable sequences.