Displayed Tree Phylogenetic Network Model
- Displayed tree phylogenetic network model is a framework that interprets rooted phylogenetic networks as collections of trees obtained by resolving reticulations.
- It integrates graph theory, combinatorics, and statistical phylogenetics to address containment, comparison, and optimization problems across different network classes.
- The model supports practical applications such as network reconstruction, parsimony and likelihood inference, and computational complexity analysis in evolutionary studies.
The displayed tree phylogenetic network model interprets a phylogenetic network through the rooted phylogenetic trees that the network displays after resolving reticulations. In the rooted setting, a network induces a display set, written either or , and this set becomes the basic semantic object for containment, comparison, reconstruction, counting, and inference problems (Kelk et al., 2017, Döcker et al., 2019, Bordewich et al., 2024). The model sits at the intersection of graph theory, combinatorics, and statistical phylogenetics: it supports exact questions such as whether a given tree is displayed, comparative questions about equality or overlap of two display sets, and statistical questions that optimize parsimony or likelihood over displayed trees rather than over unrestricted trees (Iersel et al., 2010, Sullivant, 30 Jul 2025).
1. Formal model and display relation
In the standard rooted binary formulation, a phylogenetic network on a taxon set is a rooted acyclic digraph whose root has out-degree two, whose leaves are exactly the taxa in , and whose remaining internal vertices are either tree vertices of indegree $1$, outdegree $2$, or reticulations of indegree $2$, outdegree $1$ (Kelk et al., 2017, Döcker et al., 2019). A rooted phylogenetic tree is the special case with no reticulations.
A rooted phylogenetic tree is displayed by a network 0 if, after deleting edges and vertices and then suppressing vertices of indegree 1 and outdegree 2, one obtains 3; equivalently, in the binary case one may choose exactly one incoming reticulation edge at each reticulation, forming a switching, and then suppress the resulting degree-4 vertices (Kelk et al., 2017, Döcker et al., 2019). This switching equivalence is fundamental because it turns the model into a finite combinatorial family of rooted trees. The displayed-tree set is therefore
5
The same display language extends beyond single trees. Two-network comparison problems ask whether 6, whether 7, or whether 8 (Döcker et al., 2019). Some work also uses a more general graph-embedding notion for displayed subnetworks: if 9, then 0 displays 1 when each arc of 2 is represented by a directed path in 3 whose interiors are pairwise vertex-disjoint (Francis et al., 2020). By specialization, displayed trees fit into that broader embedding framework.
2. Structural classes and tree-based interpretations
Displayed-tree behavior depends strongly on the structural subclass of rooted network under consideration. Two central classes are tree-child and normal networks. A tree-child network is one in which every non-leaf vertex has a child that is either a leaf or a tree vertex; equivalently, every vertex is visible in the sense that some root-to-leaf path structure makes it unavoidable on at least one lineage (Döcker et al., 2019, Francis et al., 2020). A normal network is a tree-child network with no shortcut arc 4 for which another directed path from 5 to 6 already exists (Francis et al., 2020). The inclusion chain emphasized in normalization work is
7
Tree-based networks are closely related to displayed-tree models but are not identical to them. In rooted work, a network is tree-based if it has a rooted spanning tree whose leaves are all in 8; that spanning tree is the network’s base tree (Francis et al., 2016, Zhang, 2015). Louxin Zhang’s characterization of rooted tree-basedness uses a bipartite matching condition and yields universal tree-based networks, meaning networks on 9 that have every phylogenetic tree on 0 as a base (Zhang, 2015). Rooted tree-basedness also admits equivalent formulations in terms of path partitions, antichains, and matchings in an associated bipartite graph 1 (Francis et al., 2016).
The distinction between displayed trees and base trees is sharper in unrooted work. There, a network is tree-based if it contains a spanning tree whose leaf set is exactly 2, and a base tree is obtained by suppressing degree-2 vertices in that support tree; every base tree is displayed, but not every displayed tree is a base-tree (Francis et al., 2017). This distinction matters because the displayed-tree phylogenetic network model is fundamentally about the whole display set, whereas tree-based frameworks ask whether one spanning tree-like scaffold underlies the network at all.
3. Decision, comparison, and optimization problems
The algorithmic landscape of displayed-tree problems is sharply stratified by network class and by whether one asks about one tree, two networks, or an optimization over all displayed trees (Iersel et al., 2010, Kelk et al., 2017, Döcker et al., 2019).
| Problem | Setting | Complexity |
|---|---|---|
| Tree Containment | normal, binary tree-child, binary level-3 with fixed 4 | polynomial-time |
| Common-Tree-Containment | two temporal normal networks | NP-complete |
| Display-Set-Containment | general networks; also hard with a temporal tree-child caterpillar on one side | 5-complete |
| Display-Set-Equivalence | general networks | 6-complete |
| Most parsimonious displayed tree | binary level-1, two states | NP-hard |
| Most parsimonious displayed tree with gaps | binary level-1 | APX-hard |
| Most likely displayed tree | binary level-1, Cavender–Farris model | NP-hard |
For single-tree containment, polynomial-time algorithms exist on normal and binary tree-child networks. In the normal case, LocateNormal decides Tree Containment and, when successful, finds the unique subtree of the network that is a subdivision of the query tree (Iersel et al., 2010). In the binary tree-child case, LocateTreeChild remains polynomial-time but must distinguish four reticulation types, including configurations with shortcuts and cut-edge decompositions (Iersel et al., 2010).
For two-network comparison, complexity rises substantially. Common-Tree-Containment is NP-complete even for two temporal normal networks, and display-set containment and display-set equivalence are 7-complete (Döcker et al., 2019). Thus moving from “does one network display one tree?” to “how do two display sets relate?” changes the problem class qualitatively.
Optimization over displayed trees is already hard in extremely restricted networks. In the “single-tree-for-the-whole-alignment” model of Nakhleh et al., the most parsimonious displayed-tree problem is
8
not a site-wise minimization over separate displayed trees (Kelk et al., 2017). Under this model, finding the most parsimonious or most likely displayed tree is NP-hard even on binary level-1 networks with only two character states, and with gaps the parsimony problem becomes APX-hard (Kelk et al., 2017). The same paper nonetheless argues that naïve enumeration of at most 9 displayed trees can be competitive in practice when the reticulation count 0 is moderate.
4. Expressiveness, reconstruction, and extremal display-set theory
One major line of work asks what collections of trees can be realized as display sets, and when a display set determines the network uniquely. For binary normal networks, the display set is a complete invariant: no two binary normal networks have the same display set, a result due to Willson and used as the starting point for a polynomial-time reconstruction algorithm in “When is a set of phylogenetic trees displayed by a normal network?” (Bordewich et al., 2024). That paper gives the recursive algorithm Display Set Compatibility, which reconstructs the unique binary normal network whose display set is exactly a given collection 1, when such a network exists, in
2
time (Bordewich et al., 2024). It also characterizes displayability by normal networks via a restricted cherry-picking formalism: 3 is normal compatible if and only if it admits a normal cherry-picking sequence, and the minimum number of reticulations over all normal networks displaying 4 equals the minimum sequence weight (Bordewich et al., 2024).
Expressive power differs sharply between normal and tree-child classes. Any two rooted phylogenetic trees are normal compatible, but this does not extend to arbitrary larger collections; the three binary trees on leaf set 5 are not normal compatible (Bordewich et al., 2024). In contrast, any two phylogenetic trees can always be simultaneously displayed by some binary tree-child network, and any finite collection of binary trees can always be simultaneously displayed in a suitable non-binary tree-child network whose nonleaf vertices are of either indegree 6, outdegree 7, or indegree at least 8, outdegree 9 (Wu et al., 2022).
Displayed-tree counting has produced both upper and lower extremal results. For normal networks with $1$0 reticulations, the display-set size is exactly $1$1 (Bordewich et al., 2024, Semple et al., 19 Aug 2025). For binary tree-child networks on $1$2 leaves, the maximum number of distinct rooted binary phylogenetic $1$3-trees is $1$4, this bound is sharp for every $1$5, and in the extremal case exactly one displayed tree is duplicated and can be canonically recovered by iteratively replacing a reticulated cherry with a cherry (Murakami et al., 12 Jun 2026). At the opposite end, if a tree-child network has $1$6 reticulations and no underlying $1$7-cycles, then it displays at least $1$8 trees when $1$9 is even and at least
$2$0
when $2$1 is odd; these lower bounds are sharp, and equality characterizes the octopus networks built from $2$2-tight and $2$3-tight caterpillar-ladder components (Semple et al., 19 Aug 2025). Computing $2$4 is $2$5-complete in general, as cited in the 2026 extremal paper (Murakami et al., 12 Jun 2026).
5. Statistical formulations and graphical-model embeddings
The displayed-tree model also appears as a statistical model. In alignment-based inference, one standard formulation chooses a single displayed tree for the whole alignment and then evaluates parsimony or likelihood on that tree (Kelk et al., 2017). Under the Cavender–Farris binary symmetric model, network likelihood in this setting is not optimized over arbitrary branch lengths on arbitrary trees; rather, it is optimized over displayed trees together with branch parameters inherited from the network, with suppressed edges combined by
$2$6
A more recent development places the displayed-tree phylogenetic network model inside graphical-model theory. In “Phylogenetic network models as graphical models,” the displayed-tree model is formulated as a submodel of the DAG graphical model on the network, with each reticulation $2$7 carrying local conditional distribution
$2$8
(Sullivant, 30 Jul 2025). This viewpoint yields a local-modification theory for DAG models, shows that stacked reticulations and hidden $2$9-blobs can be distributionally invisible, proves that reticulation parameters are locally nonidentifiable with dimension loss $2$0 at an indegree-$2$1 reticulation in the $2$2-state general Markov model, and derives flattening-rank bounds such as
$2$3
when an edge cut $2$4 separates leaf sets $2$5 and $2$6 (Sullivant, 30 Jul 2025). Within this framework, the displayed-tree model is not merely a combinatorial mixture but a restricted hidden-variable DAG model.
Graphical-model methods also clarify where displayed-tree formulations stop being adequate. Belief-propagation formulations on phylogenetic networks show that many likelihood calculations traditionally written as sums over displayed trees can instead be performed directly on a clique tree or cluster graph, with complexity governed by treewidth rather than by the raw number of displayed trees (Teo et al., 2024). At the same time, the main continuous-trait models in that work are not displayed-tree mixtures: a hybrid node may depend simultaneously on all parents through a weighted-average Gaussian law rather than by selecting one parent. The same caution appears in SNaQ pseudolikelihood theory. Under the MSCN/quartet-CF model, quartet concordance factors are often weighted averages of tree CFs, but they are not simply a global displayed-tree mixture; small blobs can be CF-equivalent to trees, 2-cycles are not detectable, 3-cycles are only generically detectable under favorable sampling and remain numerically nonidentifiable, and 4-cycles and larger become detectable under stronger sampling assumptions, with the “bad diamond” exceptions for parameter identifiability (Solis-Lemus et al., 2020).
6. Simplification, summaries, and downstream applications
Displayed-tree analysis frequently requires simplification of a tangled inferred network, but simplification and displayed-tree preservation are not equivalent goals. The normalization map
$2$7
constructs from any rooted phylogenetic network a normal network on the visible vertices, obtained by taking the Hasse diagram of reachability among visible vertices and suppressing subdividing vertices (Francis et al., 2020). The normalized network $2$8 is always normal, the map is idempotent, and it preserves reachability and clusters of retained vertices. However, it is not a displayed-tree-preserving transformation: $2$9 need not be displayed by $1$0, and one explicit example has the property that the original and normalized networks share no displayed tree at all (Francis et al., 2020). Normalization is therefore a canonical structural summary, not an equivalent displayed-tree semantics.
Displayed-tree sets also support downstream quantitative summaries. In biodiversity applications, a rooted phylogenetic network $1$1 is associated with the displayed-tree multiset $1$2, and phylogenetic diversity is aggregated over that set by operators such as $1$3, $1$4, $1$5, unweighted average, inheritance-probability-weighted average, and maximum-likelihood displayed tree (Wicke et al., 2017). Because not every reticulation-edge choice necessarily yields a valid phylogenetic $1$6-tree, the displayed-tree probabilities are renormalized over valid displayed trees only. The same framework induces displayed-tree versions of the Fair Proportion Index and Shapley Value, and was implemented in the Perl package NetDiversity for inheritance-probability-independent quantities (Wicke et al., 2017). In parallel, normalization was implemented in PhyloSketch as a polynomial-time simplification tool for rooted networks (Francis et al., 2020).
A nearby but distinct research direction studies spaces of tree-based networks rather than display sets. In the unrooted tree-based setting, the space $1$7 of tier-$1$8 tree-based networks on $1$9 leaves is connected under NNI, and the all-tier space is connected under NNI together with triangle moves (Fischer et al., 2019). This is structurally relevant to displayed-tree work because tree-basedness formalizes the presence of an underlying support tree, but it remains different from the full displayed-tree model, which is governed by whole display sets rather than by the existence of one spanning tree scaffold.
The displayed tree phylogenetic network model is therefore best understood as a family of rooted-network semantics built around the display set rather than around any single backbone tree. Its core theory now includes exact combinatorial definitions, sharp complexity boundaries, reconstruction theorems for normal networks, extremal counting results for tree-child networks, graphical-model embeddings with explicit nonidentifiability phenomena, and summary procedures that may preserve clusters or visible structure without preserving displayed trees themselves (Bordewich et al., 2024, Sullivant, 30 Jul 2025, Francis et al., 2020).