Kaikoura tree theorems: computing the maximum agreement subtree
Information Processing Letters
| Hi-index | 0.04 |
In this paper, we investigate an extremal problem on binary phylogenetic trees. Given two such trees T"1 and T"2, both with leaf-set {1,2,...,n}, we are interested in the size of the largest subset S@?{1,2,...,n} of leaves in a common subtree of T"1 and T"2. We show that any two binary phylogenetic trees have a common subtree on @W(logn) leaves, thus improving on the previously known bound of @W(loglogn) due to Steel and Szekely. To achieve this improved bound, we first consider two special cases of the problem: when one of the trees is balanced or a caterpillar, we show that the largest common subtree has @W(logn) leaves. We then handle the general case by proving and applying a Ramsey-type result: that every binary tree contains either a large balanced subtree or a large caterpillar. We also show that there are constants c,@a0 such that, when both trees are balanced, they have a common subtree on cn^@a leaves. We conjecture that it is possible to take @a=1/2 in the unrooted case, and both c=1 and @a=1/2 in the rooted case.