  We study the average height of random trees generated by leaf-centric binary
tree sources as introduced by Zhang, Yang and Kieffer. A leaf-centric binary
tree source induces for every $n \geq 2$ a probability distribution on the set
of binary trees with $n$ leaves. Our results generalize a result by Devroye,
according to which the average height of a random binary search tree of size
$n$ is in $\mathcal{O}(\log n)$.
