Given a set of leaf-labeled trees with identical leaf sets, the well-known Maximum Agreement Subtree (MAST) problem consists in finding a subtree homeomorphically included in all input trees and with the largest number of leaves. MAST and its variant called Maximum Compatible Tree (MCT) are of particular interest in computational biology. This paper presents a linear-time approximation algorithm to solve the complement version of MAST, namely identifying the smallest set of leaves to remove from input trees to obtain isomorphic trees. We also present an O(n^2 + kn) algorithm to solve the complement version of MCT. For both problems, we thus achieve significantly lower running times than previously known algorithms. Fast running times are especially important in phylogenetics where large collections of trees are routinely produced.