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A fringe subtree of a rooted tree is a subtree consisting of one of the nodesand all its descendants. In this paper, we are specifically interested in thenumber of non-isomorphic trees that appear in the collection of all fringesubtrees of a binary tree. This number is analysed under two different randommodels: uniformly random binary trees and random binary search trees.  In the case of uniformly random binary trees, we show that the number ofnon-isomorphic fringe subtrees lies between $c_1n/\sqrt{\ln n}(1+o(1))$ and$c_2n/\sqrt{\ln n}(1+o(1))$ for two constants $c_1 \approx 1.0591261434$ and$c_2 \approx 1.0761505454$, both in expectation and with high probability,where $n$ denotes the size (number of leaves) of the uniformly random binarytree. A similar result is proven for random binary search trees, but the orderof magnitude is $n/\ln n$ in this case.  Our proof technique can also be used to strengthen known results on thenumber of distinct fringe subtrees (distinct in the sense of ordered trees).This quantity is of the same order of magnitude in both cases, but withslightly different constants in the upper and lower bounds.

On the Collection of Fringe Subtrees in Random Binary Trees