Maximum Acyclic Agreement Forest (MAAF)

• MAAF is a minimum acyclic agreement forest that partitions two rooted phylogenetic trees into components reflecting reticulate evolutionary relationships.
• It links tree comparison metrics by equating the forest size with both rooted subtree prune-and-regraft distance and hybridization number, ensuring no ancestral conflicts.
• Algorithmic approaches, from exact methods like allMAAFs to approximation techniques, enable practical computation of MAAF in diverse phylogenetic settings.

Searching arXiv for recent and foundational papers on Maximum Acyclic Agreement Forest. {"query":"Maximum Acyclic Agreement Forest MAAF phylogenetic arXiv", "max_results": 10} {"query":"allMAAFs maximum acyclic agreement forest arXiv", "max_results": 10} {"query":"Ordered Leaf Attachment maximum acyclic agreement forest arXiv 2025", "max_results": 5} A maximum acyclic agreement forest (MAAF) is an acyclic agreement forest with the minimum possible number of components for a pair of rooted phylogenetic trees. In the rooted setting, it is the combinatorial object that connects tree comparison to reticulate evolution: for two rooted binary trees, MAAF characterizes minimum hybridization complexity, while closely related agreement-forest constructions characterize rooted subtree prune-and-regraft distance. Subsequent work extended the framework to multiple rooted binary trees, to rooted nonbinary trees, and, by abstraction rather than exact equivalence, to rooted phylogenetic networks (Whidden et al., 2011, Albrecht, 2014, Klawitter, 2018).

1. Formal definition and acyclicity criterion

In the rooted binary setting, an agreement forest for two rooted binary phylogenetic trees $S$ and $T$ on the same taxa set $X$ is a collection

$\mathcal{F}=\{F_\rho,F_1,\dots,F_k\}$

such that the label sets partition $X\cup\{\rho\}$, each component satisfies

$F_i \cong S|_{(F_i)} \cong T|_{(F_i)},$

and the induced subtrees $S((F_i))$ and $T((F_i))$ are vertex-disjoint within each input tree (Scornavacca et al., 2011). Equivalent formulations describe agreement forests as the result of deleting edges from both trees and suppressing degree-2 unlabeled vertices (Whidden et al., 2011).

Acyclicity is imposed through a directed dependency graph on the forest components. In one formulation, $AG(S,T,\mathcal{F})$ has vertex set $\mathcal{F}$, and an arc $T$0 whenever $T$1 and either the root of $T$2 is an ancestor of the root of $T$3 in $T$4, or the corresponding ancestry holds in $T$5 (Scornavacca et al., 2011). Other papers use closely related objects: a directed graph $T$6 on components (Mukhopadhyay et al., 2014), the inheritance graph $T$7 (Iersel et al., 2012), or a modified ancestor-descendant graph $T$8 (Albrecht, 2015). In each case, the forest is acyclic precisely when this directed graph has no directed cycle.

A MAAF is then an acyclic agreement forest with minimum cardinality. The terminology is historically counterintuitive but standard: “maximum” refers to maximum agreement, while the optimization criterion is minimum number of components. In nonbinary formulations, the same idea is retained, except that each component is required to be a refinement of the restricted trees rather than literally identical to both restrictions (Albrecht, 2015).

2. Relationship to rooted SPR distance and hybridization number

The importance of MAAF is inseparable from two classical identities. For rooted binary trees, the size of a maximum agreement forest satisfies

$T$9

so MAF characterizes rooted subtree prune-and-regraft distance (Mukhopadhyay et al., 2014, Whidden et al., 2011). The acyclic variant is the hybridization analogue: $X$0 for the usual size convention (Mukhopadhyay et al., 2014). In the labeled-root convention $X$1, the equivalent statement is that if $X$2 is a maximum-acyclic-agreement forest for $X$3, then

$X$4

(Scornavacca et al., 2011).

The extra acyclicity condition is not a cosmetic refinement of MAF. It excludes conflicting ancestral relations among forest components, and the biological interpretation given in the literature is that it prevents “species inheriting genetic material from their own offsprings” (Scornavacca et al., 2011). This is why minimizing agreement-forest size alone does not suffice for hybridization-number inference: MAF measures rooted rearrangement disagreement, whereas MAAF measures the part of that disagreement compatible with a reticulate evolutionary history (Mukhopadhyay et al., 2014).

Counting conventions vary across papers because some formulations attach an extra marker taxon $X$5 to the root. One paper states that if a MAAF $X$6 has size $X$7, then $X$8 in the usual convention, but also notes that in its own notation with the marked root component, “if $X$9 has size $\mathcal{F}=\{F_\rho,F_1,\dots,F_k\}$0, then $\mathcal{F}=\{F_\rho,F_1,\dots,F_k\}$1” (Albrecht, 2015). The underlying equivalence is the same; only the component count convention differs.

3. Exact algorithms and complete enumeration

The fixed-parameter study of MAAF for two rooted binary trees developed along a two-phase pattern: first compute an agreement forest, then refine it to an acyclic agreement forest by breaking cycles. A notable exact algorithm branches on sibling pairs in the MAF phase and then uses an expanded cycle graph in the refinement phase. The refinement identifies essential components and exit nodes, and fixing a suitable exit node reduces the remaining acyclic-cut budget. The resulting decision algorithm for $\mathcal{F}=\{F_\rho,F_1,\dots,F_k\}$2 was improved from $\mathcal{F}=\{F_\rho,F_1,\dots,F_k\}$3 to $\mathcal{F}=\{F_\rho,F_1,\dots,F_k\}$4, and then to

$\mathcal{F}=\{F_\rho,F_1,\dots,F_k\}$5

with kernelization yielding

$\mathcal{F}=\{F_\rho,F_1,\dots,F_k\}$6

(Whidden et al., 2011).

Enumeration of all optimal solutions was addressed by the algorithm allMAAFs, the first algorithm that calculates all maximum-acyclic-agreement forests for two rooted binary phylogenetic trees on the same set of taxa (Scornavacca et al., 2011). Its recursive search is organized around common and contradicting cherries, with later reversal of cherry reductions by cherryExpansion. The main correctness statement is that

$\mathcal{F}=\{F_\rho,F_1,\dots,F_k\}$7

returns all maximum-acyclic-agreement forests for $\mathcal{F}=\{F_\rho,F_1,\dots,F_k\}$8 and $\mathcal{F}=\{F_\rho,F_1,\dots,F_k\}$9 if and only if $X\cup\{\rho\}$0. The basic worst-case running time is

$X\cup\{\rho\}$1

and, with subtree and chain reductions, the extended version allMAAFs* runs in

$X\cup\{\rho\}$2

(Scornavacca et al., 2011).

Practical performance was later improved by two modifications of allMAAFs. One batches all pendant-edge cuts for certain contradicting cherries; the other contracts common cherries without branching and defers acyclicity handling to a refinement step based on the expanded cycle graph. The asymptotic worst-case bounds do not improve, but a simulation study reported mean speedups of about $X\cup\{\rho\}$3 for the first modification and $X\cup\{\rho\}$4 for the second relative to the original, with the refinement-based method described as on average about $X\cup\{\rho\}$5 times faster than the first modified version (Albrecht, 2015).

4. Exact, approximation, and reduction-based methods beyond the basic binary pair

For two rooted phylogenetic trees that are not necessarily binary, the hybridization number can be computed in

$X\cup\{\rho\}$6

and the same bound also yields computation of a MAAF (Iersel et al., 2013). The underlying dynamic program uses clusters, ST-sets, and ST-set tree sequences. If

$X\cup\{\rho\}$7

then the key recurrence is

$X\cup\{\rho\}$8

with subproblems indexed by subsets of $X\cup\{\rho\}$9, hence at most $F_i \cong S|_{(F_i)} \cong T|_{(F_i)},$0 states (Iersel et al., 2013).

Approximation algorithms were developed in both binary multi-tree and nonbinary two-tree settings. For $F_i \cong S|_{(F_i)} \cong T|_{(F_i)},$1 rooted binary phylogenetic trees, a $F_i \cong S|_{(F_i)} \cong T|_{(F_i)},$2-approximation algorithm for MAAF was given, with the theorem

$F_i \cong S|_{(F_i)} \cong T|_{(F_i)},$3

(Mukhopadhyay et al., 2014). The cycle-breaking phase operates on roots of forest components, using preorder-numbering and interval ancestry checks to detect cyclic pairs and justify local cuts (Mukhopadhyay et al., 2014).

For nonbinary trees, the algorithmic route to MAAF is indirect. One paper proves that nonbinary MAF has a polynomial-time $F_i \cong S|_{(F_i)} \cong T|_{(F_i)},$4-approximation and an exact fixed-parameter algorithm running in

$F_i \cong S|_{(F_i)} \cong T|_{(F_i)},$5

and then shows that a $F_i \cong S|_{(F_i)} \cong T|_{(F_i)},$6-approximation for nonbinary MAF combined with a $F_i \cong S|_{(F_i)} \cong T|_{(F_i)},$7-approximation for Directed Feedback Vertex Set yields a

$F_i \cong S|_{(F_i)} \cong T|_{(F_i)},$8

-approximation for nonbinary MAAF (Iersel et al., 2012). Instantiating $F_i \cong S|_{(F_i)} \cong T|_{(F_i)},$9 gives a $S((F_i))$0-approximation, and with the best-known polynomial-time approximation for weighted DFVS the result is a polynomial-time

$S((F_i))$1

-approximation for nonbinary MAAF (Iersel et al., 2012). The structural reason is that acyclicity can be enforced by splitting a maximal agreement forest, and the minimum such splitting is reducible to a weighted DFVS instance (Iersel et al., 2012).

5. Multiple trees, nonbinary trees, and hybridization-network construction

For multiple rooted binary phylogenetic trees, maximum acyclic agreement forests remain sufficient for constructing minimum hybridization networks. The algorithm allHNetworks computes all topologically distinct relevant minimum hybridization networks by exploring all orderings of the input trees, all embedded trees of each intermediate network, all MAAFs between the chosen embedded tree and the next input tree, all acyclic orderings of each MAAF, and all valid source-target placements for each forest component (Albrecht, 2014). The crucial sufficiency result is that, for computing minimum hybridization networks for multiple rooted binary phylogenetic trees on the same set of taxa, it suffices to take only maximum acyclic agreement forests into account (Albrecht, 2014).

This multi-tree theory is complemented by a cluster decomposition theorem. If all trees share a common cluster $S((F_i))$2, then

$S((F_i))$3

where $S((F_i))$4 is the subproblem restricted to $S((F_i))$5 and $S((F_i))$6 is the contracted instance in which $S((F_i))$7 is replaced by a new taxon $S((F_i))$8 (Albrecht, 2014). The paper states that the minimum hybridization number for a set of rooted binary phylogenetic trees can be computed by solving subproblems referring to common clusters of the input trees (Albrecht, 2014).

The nonbinary case required a separate exact framework. The algorithm allMulMAAFs computes all relevant nonbinary maximum acyclic agreement forests for two rooted nonbinary phylogenetic trees on the same taxa set (Albrecht, 2015). Here “relevant” means that no edge can be contracted while preserving the agreement-forest property. The method extends cherry-based branching to multifurcating trees and then uses an expanded ancestor-descendant graph $S((F_i))$9 with hybrid edges, mappings $T((F_i))$0 and $T((F_i))$1, and exit nodes whose fixing destroys cycles (Albrecht, 2015). The worst-case running time is

$T((F_i))$2

with the paper noting an optimization from Li’s thesis that can reduce the refinement factor to roughly $T((F_i))$3 (Albrecht, 2015). The implementation in Hybroscale reproduced hybridization numbers obtained by Dendroscope and TerminusEst on the reported test scenarios (Albrecht, 2015).

6. Network generalizations, limitations, and newer representations

The rooted-tree MAAF framework motivated a network-level analogue, but the exact correspondence does not survive unchanged. For rooted binary phylogenetic networks, maximum agreement graphs generalize agreement forests by adding disagreement edges that account for reticulation-number differences. If a maximum agreement graph has $T((F_i))$4 sprouts and reticulation-number difference $T((F_i))$5, the agreement distance is

$T((F_i))$6

and this is a metric on rooted binary phylogenetic networks (Klawitter, 2018). On trees, the framework reduces to the classical picture: the agreement distance on $T((F_i))$7 is equivalent to the rSPR-distance (Klawitter, 2018). On networks, however, it is only a bounded surrogate: $T((F_i))$8

$T((F_i))$9

and therefore

$AG(S,T,\mathcal{F})$0

(Klawitter, 2018). The paper gives an explicit example with

$AG(S,T,\mathcal{F})$1

showing that MAAF-style agreement structures no longer exactly characterize shortest rearrangement sequences once reticulations are intrinsic to the objects being compared (Klawitter, 2018).

A more recent line of work links MAAF to vector encodings. Ordered Leaf Attachment (OLA) vectors encode a rooted binary phylogenetic tree as an integer vector of length $AG(S,T,\mathcal{F})$2 relative to a leaf ordering, with both encoding and decoding in linear time (Markin et al., 19 Sep 2025). The corrected OLA distance $AG(S,T,\mathcal{F})$3 is defined by propagating mismatches through attachment positions, and the main theorem states that if $AG(S,T,\mathcal{F})$4, then

$AG(S,T,\mathcal{F})$5

(Markin et al., 19 Sep 2025). The same paper shows that a MAAF can be reconstructed directly from optimal OLA vectors and extends the result to multifurcated trees via an

$AG(S,T,\mathcal{F})$6

algorithm that optimally resolves a set of multifurcated trees given a leaf ordering (Markin et al., 19 Sep 2025). This establishes a representation-theoretic view of MAAF in which reticulation events are recoverable from optimal order-dependent encodings rather than only from explicit forest search.

A persistent limitation across this literature is that results for ordinary agreement forests do not automatically transfer to the acyclic case. Recent kernelization for multiple-tree MAF gives kernels of size

$AG(S,T,\mathcal{F})$7

for rooted and unrooted MAF, but it explicitly does not define acyclicity or prove that its reductions preserve the extra condition required for MAAF (Kelk et al., 25 Feb 2026). That distinction reflects a broader methodological boundary: MAAF is not simply MAF with an additional check, but the rooted, ancestry-sensitive form of agreement-forest theory that makes hybridization-number inference possible.