  Phylogenetic trees are leaf-labelled trees, where the leaves correspond to
extant species (taxa), and the internal vertices represent ancestral species.
The evolutionary history of a set of species can be explained by more than one
phylogenetic tree, giving rise to the problem of comparing phylogenetic trees
for similarity. Various distance metrics, like the subtree prune-and-regraft
(SPR), tree bisection reconnection (TBR) and nearest neighbour interchange
(NNI) have been proposed to capture this similarity. The distance between two
phylogenetic trees can also be measured by the size of a Maximum Agreement
Forest (MAF) on these trees, as it has been shown that the rooted subtree
prune-and-regraft distance is 1 less than the size of a MAF. Since computing a
MAF of minimum size is an NP-hard problem, approximation algorithms are of
interest. Recently, it has been shown that the MAF on k(>=2) trees can be
approximated to within a factor of 8. In this paper, we improve this ratio to
3. For certain species, however, the evolutionary history is not completely
tree-like. Due to reticulate evolution two gene trees, though related, appear
different, making a phylogenetic network a more appropriate representation of
reticulate evolution. A phylogenetic network contains hybrid nodes for the
species evolved from two parents. The number of such nodes is its hybridization
number. It has been shown that this number is 1 less than the size of a Maximum
Acyclic Agreement Forest (MAAF). We show that the MAAF for k(>= 2) phylogenetic
trees can be approximated to within a factor of 3.
