We study the average number of distinct fringe subtrees in 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. We generalize a result by Flajolet, Gourdon, Martinez and Devroye, according to which the average number of distinct fringe subtrees in a random binary search tree of size $n$ is in $Θ(n/\log n)$, as well as a result by Flajolet, Sipala and Steayert, according to which the number of distinct fringe subtrees in a uniformly random binary tree of size $n$ is in $Θ(n/\sqrt{\log n})$.