Bounds on the Expected Size of the Maximum Agreement Subtree for a Given Tree Shape

Abstract

We show that the expected size of the maximum agreement subtree of two n-leaf trees, uniformly random among all trees with the shape, is Θ( √ n). To derive the lower bound, we prove a global structural result on a decomposition of rooted binary trees into subgroups of leaves called blobs. To obtain the upper bound, we generalize a first moment argument from [1] for random tree distributions that are exchangeable and not necessarily sampling consistent.

Expected number of induced subtrees shared by two independent copies of a random tree

“…Bernstein, Ho, Long, Steel, St. John, and Sullivant [6] established a qualitatively similar upper bound O(n 1/2 ) for the likely size of a common induced subtree in a harder case of Yule-Harding tree, again relying on sampling consistency of this tree model. Recently Misra and Sullivant [19] were able to prove the estimate Θ(n 1/2 ) for the case when two independent binary trees with n labelled leaves are obtained by selecting independently, and uniformly at random, two leaf-labelings of the same unlabelled tree. Using the classic results on the length of the longest increasing subsequence in the uniformly random permutation, the authors of [6] established a first power-law lower bound cn 1/8 for the likely size of the common induced subtree in the case of the uniform rooted binary tree, and a lower bound cn a−o (1) , a = 0.344 .…”

Trees in Many Contexts

“…Simulations suggest that the true exponent is close to 1 2 ; see [2]. For random trees of the same shape (but different labels), it was shown recently [8] that the order of the expected size is Θ(√𝑛).…”

On the Largest Common Subtree of Random Leaf-Labeled Binary Trees

“…and so q n,K ≥ n β−1 , giving the desired lower bound via (11) and ( 9). The argument is formalized in the next two sections.…”

“…This question (for the uniform model and some other models) has already been considered in several papers 3 , most recently in [4,11], and the relevant known results 4 are as follows.…”

“…• In the alternate model where the two trees have the same shape (given the first uniform random tree, obtain the second by making a uniform random permutation of leaf-labels), [11] shows that the order of magnitude is exactly n 1/2 .…”