Zhang, Yang and Kieffer defined depth-centric binary tree sources as follows: a tree of depth $n$ (here, "depth" means the greatest distance from the root to a leaf) is constructed by the following recursive algorithm: pick a random pair $(i,j)$ such that $\max(i,j) = n-1$ according to a prescribed probability distribution $\sigma$, then recursively construct random trees of depth $i$ and $j$ respectively that are attached to a common root as left and right branch.

We discuss statistical properties of random trees from a depth-centric source for natural choices of the probability distribution $\sigma$, encountering asymptotic behaviour that is fairly unusual in the context of trees.