Maximum agreement subtree problem

Problem

The maximum agreement subtree problem is any of several closely related problems in graph theory and computer science. In all of these problems one is given a collection of trees $T_1,\ldots, T_m$ each containing $n$ leaves. The leaves of these trees are given labels from some set $L$ with $|L|=n$ so that no pair of leaves in the same tree sharing the same label, within the same tree the labelling for each leaf is distinct. In this problem one would like to find the largest subset $L'\subset L$ such that the minimal spanning subtrees containing the leaves in $L'$, of $T_1\mid S,\ldots, T_m\mid S$ are the "same" while preserving the labelling.