MDTree: A Masked Dynamic Autoregressive Model for Phylogenetic Inference
Authors: Zelin Zang
ChenRui Duan
Siyuan Li
Jinlin Wu
BingoWing-Kuen Ling
Fuji Yang
Jiebo Luo
Zhen Lei
Stan Z. Li
Venue: Accepted by TMLR ()

Abstract: Phylogenetic tree inference requires optimizing both branch lengths and topologies, yet traditional MCMC-based methods suffer from slow convergence and high computational cost. Recent deep learning approaches improve scalability but remain constrained: Bayesian models are computationally intensive, autoregressive methods depend on fixed species orders, and flow-based models underutilize genomic signals. Fixed-order autoregression introduces an inductive bias misaligned with evolutionary proximity: early misplacements distort subsequent attachment probabilities and compound topology errors (exposure bias). Absent sequence-informed priors, the posterior over the super-exponential topology space remains diffuse and multimodal, yielding high-variance gradients and sluggish convergence for both MCMC proposals and neural samplers.
We propose MDTree, a masked dynamic autoregressive framework that integrates genomic priors into a Dynamic Ordering Network to learn biologically informed node sequences. A dynamic masking mechanism further enables parallel node insertion, improving efficiency without sacrificing accuracy. Experiments on standard benchmarks demonstrate that MDTree outperforms existing methods in accuracy and runtime while producing biologically coherent phylogenies, providing a scalable solution for large-scale evolutionary analysis.

Published in Transactions on Machine Learning Research (12/2025)
MDTree: A Masked Dynamic Autoregressive Model for Phy- logenetic Inference
zangzelin Srggzelin@westakke.. edu.co Centre for Artificial Intelligence and Robotics (CAIR), Hong Kong Institute of Science and Innovation (HKTST) School of Engineering, Westlake University
Zelin Zang"
Chenrui Duan"
Chenruiduan@urestlkke...e
School Engineering, Westlake University
Siyuan Li
signankke@urrttiakeeee.cc
School of Engincering, Westlake University
Jinlin Wu
finlivenu@@kiiii-coo www.:eiiiisisiis......i. org.kk.k..............kkk...k... Centre for Artificial Intelligence and Robotics (CAIR), Hong Kong Instibute of Science and Innovation (HKIST)
wkling Center for the Integrated Circuits and Artificial Intelligence, Tsientang Institute for Advanced Study, Zhejjang 310024
HongoWing-King Ling
Fuji Yang
fyyngg@tias.oc.co
Center for the Integrated Circuits and Artificial Intelligence, Tsientang Institude for Advanced Study
Jiebo Luo
jiuo Wuo@hhisi-cas.org
Hong Kong Institute of Science and Innovation (HKISI)
Zhen Leit
Chen.eci@iaa ac..rr
Centre for Artificial Intelligence and Robotics (CAIR), Hong Kong Institute of Science and Innovation (HKISI) Institute of Automation, Chinese Academy of Sciences (CASIA) School of Artificial Intelligence, University of Chinese Academy of Sciences (UCAS)
Stan Z. Li
tta...lli@westllle edu.cn
School of Engineering, Westlake University, Hangshou, Zhejiang 310030, China
Reviewed on OpenReview: /ttps:///ppnrevveeet net/ forum? id- dTSptQNygu
Abstract
Phylogenetic tree inference requires optimizing both branch lengths and topologies, yet tra- ditional MCMC-basod methods suffer from slow convergence and high computational cost. Recent. deep learning approaches improve scalability but remain constrained: Bayesian mod- els are computationally intensive, autoregressive methods depend on fixed species orders, and flow-based models underutilize genomic signals. Fixed-order autoregression introducos an inductive bias misaligned with evolutionary proximity: early misplacements distort sub- sequent altachment probabilities and compound topology errors (exposure bias). Absent sequence-informed priors, the posterior over the super-exponential topology space remains diffusc and multimodal, yielding high-variance gradicnts and sluggish convergence for both MCMC proposals and neural samplers. We propose MDTree, masked dynamic autore- gressive framework that integrates genomic priors into a Dynamic Ordering Network to learn biologically informed node sequences. dynamic masking mechanism further enables parallel node insertion, improving efficiency without sacrificing accuracy. Experiments on
"Both authors contributed equally this research.
i "Correppondig author.

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Published in Transactions on Machine Learning Research (12/2025)
(slow)
( Sa
Sa
Sd
S
S So
Sp
Deep Autoregressive Model
(slow)
Sa
Tree Construction Branch Leaming
S
sc
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Dynamic Ordering Network
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(slow)
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Sd
Se
S So
Se
Se
+2 (faster)
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(a) Fixed-order Autoregressive Model (ARTree)
(b) MDTree (Ours)
Figure 1: Comparison between fixed-order autoregressive generation (ARTree) and our dynamic-order method (MDTreee) (a) Fixed-order ARTree adds species sequentially according to a predefined order, ignoring biological priors. This leads to suboptimal intermediate structures and slower generation, as only one leaf node is added at each step. (b) MDTree (ours) employs Dynamic Ordering Network to determine biologically-infrmmee insertion order based on genomic features, enabling related species to be clustered earlier. The Tree Construction & Branch Learning module further supports parallel insertion of multiple nodes, achieving faster generation and more phylogenctically consistent topologies.
standard benchmarks demonstrate that MDTree outperforms existing methods in accuracy and runtime while producing biologically coherent phylogenies, providing a scalable solution for large-scale evolutionary analysis.
1 Introduction
Phylogenetic trees are fundamental for revealing evolutionary relationships, enabling lineage tracing from common ancestors to present-day organisms using DNA or protein sequences (Brocchieri 2001 Munjal al., 2019h They underpin studies in taxonomy, evolutionary biology, and medicine, offering insights into species origins, biodiversity, and the cvolutionary trajectorics of pathogens and cancer cells (Hugenholtz et al. 2021
Tummers & Green 2022). Accurate and efficient inference has high practical value: in pathogen source trac-
ing, it supports timely outbreak interventions [Biek et al. 2015); ; in cancer evolution analysis (Fimereli et al. 2022h, it revcals clonal architecture and treatment resistance; and in biodiversity conservation (Theissinger et al. 2023), it enables large-scale, automated species relationship reconstruction. These applications under- score both the scientific and societal signíficance of phylogenetic modeling. Yet, the surge of genomic data and the combinatorial growth of tree topologies pose major computational challenges, calling for scalable and accurate new methods.
Traditional statistical frameworks, notably Maximum Likelihood Estimation (MLF)]2quierdoo Tzquierdo-Carrasco et al. 2011 Solís-Lemus & Ané 2016) and Bayesian Inference (BI) via Markov Chain Monte Carlo (ACMCC]Zhaaaa) et al.] 2018 Wang et al., 2020). have long underpinned phylogenetic inference. Yet, with increasing taxa, they encounter severe computational bottlenecks: the space of unrooted bifurcating topolo- gics grows super-oxponentially as (2N-5)!!, 5)!!, while the joint optimization of continuous branch lengths and discrete topologies further compounds complexity.
Leveraging deep learning, breakthroughs in phylogenetic inference have burst onto the scene, addressing long-standing computational challenges in the field (Nesterenko et al. 2022 Smith & Hahn 2023 Tang
et al. 2024). Research eflorts primarily follow two main directions: representation learning on known tree structures and generative models. The former, exemplified by VBPI-GNN (Zhang 2023). optimizes perfor- mance based on predefined topologies but struggles when the topology unknown and both topology and branch lengths must be inferred. These methods also underutilize evolutionary information from biological sequences, impacting accuracy and flexibility (Penny 2004h. On the other hand, generative models, which
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infer tree structures directly from data, can be further divided into three types: Bayesian generative mod- els (e-g., Geophy (Mimori & Hamada, 2024h) leverage probabilistic frameworks to capture uncertainty but are computationally intensive; autoregressive models (e.g., ARTree (Xie & Zhang 2024h) sequentially add nodes, offering flexibility yet relying on predefined orders that overlook true cvolutionary relationships, while their stepwise nature leads to inefficiency for large dalasets (Razavi el al. 2019 Lastly, Generative Flow Networks (GFNs) (e.g., PhyloGFN (Zhou et al.) 2024)) provide greater flexibility by exploring multimodal posterior distributions but still struggle to fully integrate evolutionary signals, impacting the accuracy of inferred trees. Therefore, few methods have achieved these goals simultaneously.
To overcome these limitations, we focus on core question: how can biological priors effectively guide node addition to improve phylogenetic inference ac- curacy? As shown in Fig. _ classical autoregressive
20
D3
ARTree(SOTA) MDTree(Ours)
methods (Fig. 2 rely on fixed orders (o.g., lexi- cographical), overlooking evolutionary relationships and often producing inaccurate trees ! Our method (Fig. p)) learns evolutionarily meaningful node or- ders, ensuring species like reptiles, birds, and mam-
Accuracy (Negative Loss
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mals are added in line with their ancestry. This improves the accuracy and biological relevance of generated trees by prioritizing species with closer
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common ancestors.
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Specifically, adopt a dynamic autoregres- sive generation paradigm, where both the or- der of node additions and their insertion positions are learned from genomic sequences instead of be- ing fixed advance. This paradigm is instanti- ated by our Masked Dynamic Autoregressive Model (MDTreee), which integrates Diffusion Ordering Network (DON) to learn biologically informed or- ders directly from sequence data via an absorbing diffusson model (Bond-Taylor et al. 2021 mitigating the limitations of fixed or random orders. By combining the strengths of Graph Neural Networks and Language Models (LMs), MDTree captures intricate genomic relationships while modeling complex tree structures. A Dynamic Masking Mechanism enables parallel node processing, improving efficiency. Lastly, we employ a dual-pass tree traversal strategy for branch length estimation and use the LAX model (Grathwohl et al. 2017 to reduce variance in discrete sampling for stabilizing optimization and enhancing convergence.
Runtime (Seconds, Lower Better)
Figure 2: Runtime and node count comparison between MDTree and ARTree. Evaluation is con- ducted on eight benchmarks under two optimization settings (log-scalo).
Experiments on phylogenetic benchmarks show that MDTree outperforms existing methods in accuracy and efficiency. Empirical analysis of Angiosperms353 (Zuntini et al. 2024 further demonstrates its ability to recover evolutionary lineages, including Rosaceae and Moraceae, suggesting broader biological applications. In summary, our contributions are:
A novel dynamic autoregressive generation paradigm for phylogenetic inference: We leverage a generation strategy that dynamically learns node order and insertion positions from genomic sequence data, improving the accuracy and biological relevance of inferred trees.
An innovative methodology: We propose MDTree, which integrates a Diffusion Ordering Net- work for biologically informed node orders, combines genomic Language Models with dual-pass traversal for precise tree generation, and employs dynamic masking mechanism for efficient paral- lel processing.
Strong experimental validation: Comprehensive experiments validate that MDTrec achieves state-of-the-art performance. Visualizations from real-world Angiosperm datasets further confirm the biological relevance and interpretability of the generated trees.
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Published in Transactions on Machine Learning Research (12/2025)
2 Related Works
Phylogenctic inference methods are generally categorized into traditional and deep learning- based ap- proaches; each is further divided into graph structure generation and represenlation models. For details on background, please see Appendix A
Traditional Methods rely on predefined evolutionary models and statistical inference. Graph Structure Generalion Models: MrBayes (Ronquist et. al. 2012) utilizes Bayesian inference to generate trees but strug- gles with high-dimensional combinatorial spaces, requiring large sample sizes for accuracy. VaiPhy (Kop- tagel ct al. 2022) combines SLANTIS sampling strategy (Diaconis 2019) with biological models (c.g.. JC model (Munro 2 to estimate branch lengths and generate accurate tree structures. Graph Struc- ture Representation Models: SBN (Zhang & Matsen IV. 2018a) models the probability distribution of tree topologies from existing trees, focusing on subsplit relationships without directly estimating branch lengths. VBPI (Zhang Matsen IV 2018b) extends SBNs to estimate posterior distributions and optimize branch lengths through variational inference.
Deep Learning-baser based Methods offer more flexible and scalable solutions. Graph Structure Genera- tion Models: (1) Bayesian Generative Models like GeoPhy (Mimori & Hamada 2024 learn latent tree representations to generate diverse topologies. (2) Autoregressive Models such as ARTree (Xie & Zhang, 2024) soquentially generate trees, well-suited for hicrarchical data.
(3) Generative Flow Networks like Phy- loGEN Zhoou el al. 2024) optimize tree generation paths using Markov decision processes. Gruph Structure Representation Models: VBPI-GNN (Zhang 2023) combines SBNs with variational inference to optimize
topology and branch lengths.
3 Background
3.1 Phylogeny and Machine Learning Applications in Biology
Phylogeny is the study of evolutionary relationships among species, aiming to infer their common ances- tors and evolutionary paths by analyzing gene or protein sequences. Phylogenetic trees are widely used in biology to represent these relationships, providing insights into species origins, biodiversity, and evolution- ary trajectorics. Phylogenctic trees play a crucial role in various applications, including pathogen source tracing, cancer evolution analysis, and biodiversity conservation. However, with the surge in genomic data, phylogenetic inference faces substantial computational challenges, especially when inferring trees for a large number of species. Traditional mothods encounter significant computational bottlenocks as the number of taxa increases, requiring more time and computational resources.
Challenges in Traditional Phylogenetic Inference Methods. Traditional phylogenetic inference meth- ods, such Maximum Likelihood Estimation (MLE) and Bayesian Inference (MCMC), rely exhaustive searches over large tree spaces to calculate the optimal topology. As the number of species increases, the combinatorial space of possible tree topologies grows exponentially, leading to severe computational bottle- necks. For example, Bayesian inference methods are computationally expensive and slow, particularly as the number of taxa increases. Additionally, traditional autoregressive models (c.g., ARTree) rely on predefined species orders, which often do not align with the actual evolutionary relationships, resulting in suboptimal tree structures and slower convergence.
The Role of Machine Learning in Phylogenetic Inference. In recent years, deep learning methods have shown great potential in improving phylogenetic inference by leveraging complex relationships in ge- nomic data. These methods allow for adaptive learning of node orders, which helps overcome the limitations of fixed species orders in traditional models. For instance, Generative Flow Networks (GFNs) and autore- gressive models, like ARTree, have improved the efficiency and accuracy of tree generation. However, these mothods still fail to fully incorporate biological prior knowledge, such as evolutionary relationships between species, and often do not capture the complete genomic signals, limiting their ability to provide accurate phylogenies for large datasets.
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Innovations of MDTree. To address the limitations of traditional methods, we propose MDTree,, novel dynamic autoregressive framework based on Dynamic Ordering Network (DON). MDTree dynamically learns the node addition order from genomic sequence data, instead of relying on predefined orders. This approach ensurcs that the tree construction process better reflects the truc evolutionary relationships among species, improving both accuracy and biological relevance. Furthermore, MDTree incorporates a dynamic masking mechanism that enables parallel insertion of nodes, significantly improving computational efficiency. By leveraging this method, we not only overcome computational bottlenecks but also ensure that the gener- ated phylogenetic trees are biologically consistent, making them suitable for large-scale evolutionary analysis.
3.2 Phylogenetic Posterior and Variational Inference (VI)
Variational Autoencoders (VAE) Kingma & Welling 2013) are deep generative models that learn the imput data distribution by encoding it into latent space. In this process, the encoder maps each input æ to a latent space defined by parameters: mean j and variance a. Latent variables 2 are then sampled from this distribution for data generation.
Variational Inference (VI) is employed within VAËs to handle the computational challenges of estimating marginal likelihoods of observed data. VI approximates the marginal likelihood using a variational distribu- tion (=(=)=r) to estimate the posterior. The goal of VI is to maximize the Evidence Lower Bound (ELBO), formulated as:
ELBO (lauttt))loogoo(t
(1)
The first term is the reconstruction log-likelihood, log (orl)),,,, which can be considered a decoder, i.e., the log-likelihood between the reconstructed data and the original data given the potential representation. The second term, the Kullback-Leibler (KL) divergence, quantifies the difference between the variational posterior (2/2x) and the latent prior p(2).
In the context of phylogenetic inference, VT helps to approximate the posterior distribution of tree topologies and branch lengths, which are often intractable to compute directly. By applying VAE with VI, we can efficiently infer phylogenetic structures while maintaining the biological relevance of the tree, improving both the accuracy and computational efficiency of the process.
4 Methods
4.1 Problem Formulation and Notation
A unified model that can handle both tasks must therefore (i) capture biologically meaningful topological structures, and (fi) accurately estimate continuous evolutionary distances, while being robust to limited or no supervision on the topology. Unless otherwise specified, all vector representations (e.g., ha) are treated as column vectors.
Formulation. Given set of N species sequences { and their multiple soquence alignment (MSA) A, along with genomic representations = (gi ) extracted from pretrained Genomic Language Model (e.g., DNABERT2 (Zhou et al. 2023
aim to infer phylogenetic tree that captures both its discrete topology and contimuous evolutionary distances. Formally, the tree is modcled as an unrooted binary graph == (Vr. E,), where V, denotes the set of taxa and internal nodes, and & the set of undirected edges between them. Each edge e E & is associated with a branch length be E IR,, and we denote the vector of all branch lengths L, (be : E,) R RR Our objective is to learn mapping, F (S.A,9) - (T, which jointly specifies the topology t and its associated branch lengths l--
Task Definitions. We evaluate MDTree on two complementary phylogenetic inference tasks:
Tree Topology Density Estimation (TDE). This task focuses on learning a flexible distribution qe(r) over tree topologics that captures the uncertainty inherent phylogenetic inference. The model trained maximize the marginal lieelioe:hooo::
(x)(() logp(A 7)], where p(A|T)= (AATT,e)D((e 1
(2)
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Node
Masked Node
Evolution Simulation Module
Selection & Placement Module Node Update Module
Generated Tree
Edge
Masked Edge
qF
24
V
Graph Construction Module
RGCN(.)
Select Nodes
Genetic
Species Order
U
Language
Model
6
6++1
Cxu
Position Scorer
Tree Updater
Species Phylogeny Prior
Cec
Tree
Tree
Tree
q
T
LDON
lteration
A. Dynamic Ordering Network (DON)
B. AutoRegressive Tree Construction
N(0, 7) Sampling
Branch Length Head (BLH)
{ma}
{+}
{b.}
by
(k)
Postorder Preorder Aggregation
DGCNN
Reparameterization
Aggregation
C. Branch Length Learning
Figure 3: Framework of MDTree for dynamic autoregressive tree generation. A. The Dynamic Ordering Network module utilizes pro-trained genomic LM to extract embeddings from soquenees S, guiding nodes into absorbing states in an autoregressive manner as determined by DON qo(a Go). B. The Autoregressive Tree Construction module employs parallel strategy to add multiple leaf and internal nodes simultaneously at specified positions based the order provided by DON, C. The Branch Length Learning module optimizes branch lengths through a dual-pass traversal.
In practice, we approximate the integral using a point estimate &, obtained from our branch length estimation module (Sec. 4.4), yielding p(A r) p(A t, L.). During TDE evaluation, we focus solely on topological accuracy metries (e.g., Robinson-Foulds distance) and discard the branch length estimates.
Variational Bayesian Phylogenetic Inference (VBPI). This task extends TDE by jointly modeling both topology and branch lengths through structured posterior q0,0TT, l-) (T) r). The model optimizes the Evidence Lower Bound (ELBO):
Egp [lo P(A r,,+) (T, p(),
where =(T)pp 7) with uniform topology prior and exponential branch length prior. During VBPI evaluation, we assess both topological accuracy and branch length estimation quality (e.g., using Euclidean distance in branch length space).
Model Architecture. To address the limitations of fixed node orders in prior autoregressive models (Xie & Zhang 2024h, we propose MDTree (Fig. Bi, which dynamically learns biologically informed node addition orders and insertion positions from G via Dynamic Ordering Network (DON) based on an absorbing diffusion process Austin et al. 2021). h. The learned order and context.al node embeddings jointly guide an autoregressive tree construction module with dynamic masking for parallel insertion. Finally, a dual-pass traversal refines branch lengths using both global and local structural cucs. MDTrce supports both TDE and VBPI within unified pipeline: the full architecture (DON Tree Construction + Branch Length Refinement) is used for both tasks, trained with combined objective. During evaluation, task-specific metrics are applied: for TDE, the branch length estimates é, serve as auxiliary variables to approximate the marginal likelihood, while for VBPI, they evaluated part of the posterior distribution.

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4.2 DON for Learning Biologically Informed Node Orders with Genomic Priors
The order in which species nodes are added to phylogenetic tree significantly impacts the inferred topol- ogy, especially under the constraint of binary unrooted trees. From a biological perspective, species with closer ancestry should be introduced earlier in the tree construction process to better preserve evolutionary semantics [Penny] 2004 Gregory 2008). While some recent. works show robustness to taxa orderings (Xie & "Zhang 2024h, the benefit of learning biologically informed node orders remains underexplored, Such an approach could allow the model to adaptively exploit genomic signals to produce topologies that better reflect true evolutionary relationships.
Graph Construction Module Given an input set sequences S, each 8n E is first encoded into a ge- nomic embedding ge Ed using a pretrained genomic language model (e.g., DNABERT2 Zhoou et al. 2023) injecting biological priors into subsequent ordering decisions. An initial graph 9o (V,E) is constructed based sequence similarity or known homology (species phylogeny prior), which serves as the structural backbone for contextual reasoning. The initial contextual embeddings h! de for each node v are com- puted by passing node fcatures through relational graph convolutional network (RGCN) (Schlichtkrul) al. 2018):
h = RGCN (g..
(4)
where g. is the genomic embedding of node v. The resulting set of embeddings (he JEEV integrates both sequence-level genomic information and local graph structure information encoded in E. These embeddings remain fixed throughout the subsequent absorbing diffusion process, which models node ordering through evolving selection probabilities rather than feature updates.
Evolution Simulation Module The node ordering process models the sequential absorption of nodes into masked state over N discrete time steps t == 1,2 N. At each step the model selects node from the remaining active set V, to add to the ordered sequence
(1) Transition dynamics: The discrete-time transition matrix Q' E (NN+1))(N++) controls state changes at step t, where each element Q represents the probability of transitioning from state to state j. The matrix is designed to model gradual absorption into a masked state m = N +
if j m (absorbing state)
1 - Br
if j + m (self-loop)
Q2
(5)
3e
if j = m, + m (absorption) otherwise,
0
where 3e (1 + cos m)) is monotonically decreasing schedule that controls the absorption rate, starting near at t and decreasing to N. For the base case, we define Q II++ (the identity matrix).
(2) Node selection probability: At each step t, the model computes the probability of absorbing each active node E V.. The active set compriscs all nodes not yet absorbed: V, = V fei...., 11-1), where during inference = oz (greedy selections), and during training v of (ground-truth prefix via teacher forcing). The fixed initial embedding h!!)) Rd is projected into the (N + 1)-dimensional state space via learned projection W Wrooj (N++)xxd Since the embeddings are fixed, the projected representation Z, is computed once and reused at all time steps. The selection probability vector (t) E N+1 for node v at step t is computed by normalizing the posterior:
_ (Q
2.)) O (Q'-IT zw)
where Z
W h E N+1
(6)
2 Q'zz
where represents dement-wise multiplication and -' Q" is the cumulative transition matrix up to step ( 2-1 1. The numerator is unnormalized probability vector in RN+1 and the denominator is
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scalar normalization constant ensuring Ei q [+] 1. Although the node embeddings remain fixed, the selection probabilities q
(t) cvolve over time through the transition dynamics.
(8) Node selection: During inference, starting from V1 = V, at each step i, the node is selected by comparing the absorption probabilities to the masked state:
(4) 2 = arg max 9 [m]. WE
where ( [m] denotes the probability of node v transitioning to the masked state m = N + at step t. The active node set is then updated: V++1 = V. {vz). This process continues until all nodes are absorbed at step N, producing the final node order = (uî, UG....,, uty). Since the node embeddings remain fixed throughout the ordering process, we denote them simply as (hrlvev in downstream modules (dropping the superscript (0)). These embeddings, together with the learned order o, serve as input to the subsequent tree construction and branch length estimation modules.
Ordering Supervision Module We supervise the Dynamic Ordering Network (DON) by aligning its predicted node absorption order with a reference order o = (ui , ,..., .f obtained from external phylogenetic inference tools (e.g., MrBayes). During training, the active node set at each step t determined by the ground-truth prefix as described above (i.e., V. = V {uf ....., ef..,,,__)) }), ensuring that DON learns to predict the next node conditioned on the correct absorption history. The supervision objective is the negative log-likelihood (NI.L)) of the ground-truth sequence:
LDON = Σ log where Pr(v) exp(a 1 (ml)
(8)
Σwev, exp(q 1 [m])
This objective encourages DON to assign high probability to biologically consistent construction sequences while maintaining differentiability for end-to-end training.
4.3 AutoRegressive Tree Construction with Dynamic Node Insertion
Given the biologically informed node ordering o = (Vi, us.....,fy o5)) and contextual node embeddings heev from the DON module, the autoregressive tree construction stage determines where each node should be inserted. For notational simplicity, we denote v, = of throughout this subsection. The embeddings are directly passed from DON without re-encoding, prescrving their biological semantics, while the ordering o is used in two complementary ways: (i) as explicit bias in insertion scoring to priorilize evolutionarily important nodes, and (ii) as scheduling signal for dynamic parallel insertion.
Selection & Placement Module Instead of the standard sequential insertion (one node per step), we propose dynamic parallel insertion that processes multiple nodes simultaneously while adapting the batch size over time. We initialize the tree with the first three nodes: Vpllaced {vI, U2, 23), g(0) {(11, (v2, po). (V3, po)} where po is an internal node with embedding rpo- {(,,, + hng hog). A each iteration t E {1, {1...., T} (distinct from pressag-passing steps in DON), we maintain the placed input nodes yplaced, remaining nodes R. V/ \ yplaced, current edges g(1), and embeddings for all nodes in the current tree (initially r, h, for input nodes). We select subset Ui C R, according to cosine schedule:
11(11
t-1 T T-11)
u, = SelecttNodes(RR..))) P: P==2
1 cos
π
where Pe E (0, 1] controls the fraction of remaining nodes to insert at iteration t. This schedule starts aggressive (inserting many nodes when the tree is sparse) and becomes conservative (inserting fewer nodes when the tree is complex), automatically balancing accuracy and efficiency. We terminate when R, = 0, which typically occurs in O(VN) iterations. For each node v E Ue, we compute its contextual embedding by attending to nodes already in the tree: H, MHA(h,, (Iu u E current tree)). For each edge (w1, u2) e g(t), we compute edge embedding
2 (r., Tuz) and score:
S MLP((r,, Fey Fu @ re, PE(())
(10)
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To explicitly leverage the biologically informed ordering, we introduce rank-based bias. (I,..,, NY denote the position of in o (i.e., Ranka i).
Let Rank, (u)
e
S Su + (N - (v))
(11)
where E RI a vector of ones and E 2 controls the strength of evolutionary guidance. This ensures that nodes prioritized by DON are favored during insertion, creating direct pathway from learned evolutionary patterns to tree topology. The insertion edge is sampled from 6p (nltinomial(sottmax(...
Node Update Module After sampling insertion edges fvet, we update the tree sequentially (follow- ing the order a) to avoid conflicts. For each E Ur e, (w1, w2) in the current edge set, we remove Cvr create new internal node p|u] and add edges (v, p(u)), (plv), un), (p(o), u2). The new internal node's embedding is initialized as rpiu] ( Fai + ru.). If ev was already removed by a previous insertion in this batch, we resample ey from the updated edge set. We update V/41 = ypllced Uld and repeat until all input nodes are placod. The final topology T = (V, has [V+] = 2N 2 nodes (including N input nodes and N internal nodes) and |&+| = 2N edges. The node embeddings {ru) logether with the contextual embeddings (holvev from DON, are passed to the branch length refinement module (Sec. 1.4
for metric estimation.
4.4 Dual-Pass Traversal for Branch Length Learning
The branch length learning module (Fig. 3 takes input the inferred topology T= = Er), node beddings huev, from the tree construction module, and the multiple sequence alignment A. It jointly captures global and local structural cues via iterative dual-pass traversal with progressive branch length refinement, followed by graph-based encoding and differentiable sampling.
Iterative Dual-Pass Aggregation We perform ilerations (typically K =) of dual-pass aggregation progressively refine branch length estimates. To enable tree traversal, we arbitrarily root the unrooted topology at the midpoint of edge; since the final topology remains unrooted, the choice of root does not affect the branch length estimates. At each iteration {1...., K}, we execute the following steps:
(1) Postorder aggregation propagates information bottom-up from leaves to root:
Fa: (Tui
if C())
(k) h u
GRU
1 [e(1]] Evectw) @(h (wd.(k) (hddd.(()) .(.))))
otherwise
(12)
where ra is the node embedding from tree construction (for both input nodes and internal nodes), C(u) is the child set u, (k) E R4 is the current branch length estimate (with e! = 0.1), and @ Rd x R + Rd is an MLP.
(2) Preorder aggregation propagates refined context top-down from rool leaves:
h (k),(k) 2b (h hwd,(k) hh
(M). PE((epth(()))
(13)
where vp IRd x dRd x gdow > IRd is an MLP, PE(-) E Iddr encodes the depth of node v, and booo
bwd,(k) h h
fk).
(3) Branch length refinement. We combine bidirectional features x (k) Concat(h (w). h, (k). For intermediate iterations (k K), we use Branch Length Head (BLH).a lightweight MLP, to predict ((()) = cxp(BLLHtte (K) xx (())) and set ((++)==[b for the next iteration.
Graph Encoding and Final Prediction After K iterations, we pass the final features {x] (K) hev, through L = 2 layers of graph attention networks (Veličković et al. 2017) on (Vr,E7), yielding refined representations (zihev. A separate branch length head (a dual-output MLP) then produces final log- Gaussian parameters. Specifically, the head outputs (ue,āe) = Beanch z.), and we set ae = exp(ā.) to ensure positivity. Diffcrentiable sampling is enabled via reparameterization:
be exp(Ue + de :e); where 6 NO 1) (training), ee (inference).
(14)
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Table 1: Research Questions (RQs) and their corresponding sub-questions.
How well does MDfree perform in generating tree topologies (TDE)
RQ1: Performance
and inferring branch lengths (VBPI)?
RQ2: Time Efficiency
How efficient is MDTree in reducing runtime?
How optimal MDTrre to generate a tree structure? (RQ3-1)
RQ3: Tree Quality
How diverse are the tree topologies generated by MDTree? (RQ3-2)
How consistent is the ADDTree-generann
compared MrBayes? (RQ3-3)
How does each MDTree's module affect its performance? (RQ4-2)
RQ4: Module Impact
How do koy hyper-paramcters affect MDTrec? (RQ4-2)
RQ5: Case Study
What evolutionary relationships between species does MDTree learn?
The collection L, = (be e E &,) forms the final branch length vector.
Branch Length Loss
The predicted branch lengths l, supervised by the negative log-likelihood under continuous-tine Markov chain (CTMC) substitution model (Yang) 1994)
Dien Σ log (ATMc T, Lr, ambb)
(15)
where PсTMO is computed via Felsenstein's pruning algorithm, A. denotes the e-th column of the aligument, and sub = {TT, Q} represents the substitution model parameters (nucleotide frequencies and rate matrix Q) estimated from A via. maximum likelihood Yang 1994). This iterative design progressively refines branch lengths through dual-pass aggregation, while the final roparameterized sampling enables stable gradient- based optimization.
4.5 MDTree Loss and Training
Building on the biologically informed node ordering (Sec. 1.2 topology construction (Sec. 1.3 and branch
length estimation (Sec. 4.4 MDTrree integrates these components via. Variational Bayesian Phylogenetic
Inference (VBPT).
VBPI Objective
The joint posterior (T, qe(T) r) is optimized via the ELBO:
LYBrI = -B (ootr)gs(e- [log((A] r, K-) KT((go(T) [[()) - Bxo(r) [KT(gs(e pl) r))),
(16)
where p(A| | T,,+) is the CTMC likelihood, and p(7), p(e r) are uniform and exponential priors.
Unified Objective The complete training objective combines node ordering supervision, VBPI, and aux- iliary branch length loss:
LMDTTee >DON + LVBPI Cien
(17)
with NDON = 0.1 and Auen = 0.5 (Eq. This end-to-end coupling yields phylogenies that are both topolog- ically accurate and metrically consistent.
5 Experiments
In this section, we demonstrate the effectiveness of our proposed MDTree in terms of the rescarch questions in Table i
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Table 2:
Comparison of KL divergence ()) across eight benchmark datasets with different methods. Boldface for the highest result, Underline for the second highest result of traditional methods. Results are reported as mean (standard deviation).
Dataset
DS1
DS2
DS3
DS4
DS5
DS6
DS7
DS8
(#Taxa, Sites)
(27,
(29,2520)
(36,1812)
(41,1137)
(50,378)
(50,1133)
(59,1824)
(64,1008)
Methods
Sampled Trees GT Troes
1228
43
828
33752
35407
1125
3067
2784
42
351
11505
1516877
809765
11525
82162
0.0707
0.0144
0.0554
0.0739
1.2472
0.3795
0.1531
0.3173
SHN
(0.0089)
(0.0021)
(0.0067)
(0.0095)
(0.1621)
(0.0493)
(0.0199)
(0.0412)
0.0155
0.0122
0.3539
0.5322
11.5746
10.0159
1.2765
2. 1653
MCMC- based
SRF
(0.0023)
(0.0018)
(0.0460)
(0.0692)
(1.5047)
(1.3021)
(0. (D.1659)
(0.2815)
0.6027
0.0218
0,2074
0.1952
1.3272
0.4526
0.3292
0.4149
CCD
(0.0783)
(0.0032)
(0.0270)
(0.0254)
(0.1725)
(0.0588)
(0.0428)
(0.0539)
0.0687
0.0218
0.2074
0.1952
1.3272
0.4526
0.3292
0.4149
SBN-SA
(0.0089)
(0.0032)
(0.0270)
(0.0254)
(0.1725)
(0.0588)
(0.0428)
(0.0539)
0.0136
0.0199
0.1243
0,0763
0,8599
0.3016
O 0,0488
0,1415
SBN-EM
(0.0020)
(0.0029)
(0.0162)
(0.0099)
(0.1118)
(0.0392)
(0.0063)
(0.0184)
0.0130
0.0128
0.0882
0.0637
0.8218
0.2786
0.0399
0.1236
SBN-EM-o
(0.0019)
(0.0019)
(0.0115)
(0.0083)
(0.1068)
(0.0362)
(0.0052)
(0.0161)
0.0045
0.0097
0.0548
0.0299
0.6266
0.2360
0.0191
0.0741
Structure Generation
ARTree
(0.0007)
(0.0012)
(0.0071)
(0.0039)
(0.0815)
(0.0307)
(0.0025)
(0.0096)
0.0036
0.0129
0.0446
0.0216
0.5751
0.1591
0.0169
0.0634
MDTree
(0.0005)
(0.0016)
(0.0055)
(0.0026)
(0.0690)
(0.0191)
(0.0020)
(0.0076)
5.1 Experiment Setup
Evaluation Tasks and Datasets. We assess MDTree's performance on two key tasks: TDE, which focuses optimizing tree topologies with MLL metric, and VBPI, where tree topologies and branch lengths are jointly inferred, using ELBO and MLL. These evaluations span cight diverse benchmark datascts, covering various organismss like marine animals, plants, bacteria, fungi, and eukaryotes, as outlined in Appendix C
Baselines. MDTree is compared against three primary groups of baselines: (1) MCMC- based methods (e.g., MrBayes, SBN), (2) Structure Representation methods (VBPI, VBPI-GNN), which leverage pro-generatod topologies, and (3) Structure Generation methods for Bayesian inference without pre-selected topologies. Notably, ARTree, comparable autoregrossive method like ours, is highlighted for comparison. All training details and hyperparameters are provided in Appendix D
5.2 Comparison Results on Benchmarks (RQ1)
The TDE Task. We compare the KL divergence to measure the difference between the model's generated tree topology distribution (T) and the true posterior P(T): KL(p(T)Il9((T)) = Σ, P(T) log p(r) ")(") Table 2 shows that our MDTree consistently achieves lower KL divergence across all datasets compared to MCMC- based and structure generation methods. On complex datasets such as DS5 and DS6, it outperforms ARTree and SBN, demonstrating superior scalability. Even on smaller datasets like DS1 and DS3, the performance remains competitive, highlighting the model's robustness. The comparison with ARTree underscores the advantage of autoregressive models, including ours, particularly on larger, more complex datasets.
The VBPI Task. We evaluate the VBPI task using ELBO and MLL metrics. Since direct computation of MLL is intractable, it is approximated via importance sampling. Unlike TDE, which relies on known tree topologies, VBPI evaluates the fit between model- generated tree topologies and branch lengths and the observed gene sequence data. As shown in Table g and Table 5. Tree Structure Generation methods exhibit broader applicability in MLL and ELBO metrics compared to Structure Representation methods, which are restricted by their reliance on pre-generated topologies. Our method, MDTrre,, consistently achieves the highest metries across all datascts, highlighting its enhanced capacity to approximate the postcrior distri- bution of tree topologies and branch lengths. Fig. | shows MDTree's superior stability and fast convergence in ELBO on DS1, outperforming baselines. ARTree and SBN improve later but with fluct while Geo-
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Table 3: Evaluation of MLL (1) on eight benchmark datasets. VBPI and VBPI-GNN utilize pre- generated tree topologies during training, making direct comparisons challenging. Boldface highlights the highest result, Text denotes the second highest of structure generation methods, and Text indicates the second highest of MCMC-based methods. Results reported as mcan (standard deviation).
Dataset
DS1
DS2
DS3
DS4
DS5
DS6
DS7
DS8
Methods
(# Taxa, #Stess)
(27,1949)
(29,2520)
(36,1812)
(41,1137)
(50,378)
(50,1133)
(8941
(64,1008)
MCMC- based
MrBayes
-7108.42 42
-26367.57
-33735.44
-13330.44
-8214.51
-6724.07
-37332.76
-8649.88
(0.18)
(0.48)
(0.50)
(0.54)
(0.28)
(0.86)
(2.42)
(1.75)
7108 41
26367.71
33735.08
13329.9
8214.62
6724.37
37331.97 .997
8650.64
SBN
(0.15)
(0.08)
(0.09)
(0.20)
(0.40)
(0.43)
(0.28)
(0.50)
-7008.4
26367.72
33735.10
13329.94
8214.61
6724.34
-3733.00
8650.63
Structure
VEPI
(0.10)
(0.12)
(0.11)
(0.31)
(0.67)
(0.68)
(0.43)
(0.55)
Representation
-7108.41
-26367.73
-33735.12
-13329.94
-8214.64
-724.37
-37332.01
-650.65
VBPI-GNN
(0.14)
(0.07)
(0.09)
(U.19)
(0.38)
(0.40)
(0.12)
(0.45)
-7108.41 41
-26367.71
-3733 09
-13329 94
-214.599
-6724.37
-37331.95
-650..0
ARTree
(0.19)
(0.07)
(0.09)
(0.17)
(0.34)
(0.46)
(0.27)
(0.18)
phi-CSMC
-7290.36
-30568.4
-33798.06
-13582.24
8367.51
7013.83
NA
99209.1
(7.23)
(31.34)
(6.62)
(35.08)
(8.87)
(16.99)
(18.03)
-711155
-26379 48
-33757.79
-13342.71
-8240.87
-6735.14
-37377.86
-8663351
Structure Generation
GeoPhy
(0.07)
(11.60)
(8.07)
(1.61)
(9.80)
(2.64)
(29.48)
(6.85)
-7116.09
-26368.51
33735.85
13337.42
-233.89
-673.99
33755.96
-8600.8 48
GeoPhy LOO(3)
(10.67)
(0.12)
(0.12)
(1.32)
(6.63)
(1.13)
(13.06)
(0.78)
7108.95
26368.90
33735.60
13331.8 .83
8215.15
6730.68
33359.96
8654.76
PhyloGFN
(0.06)
(0.28)
(0.35)
(0.19)
(0:20)
(0.54)
(1.14)
(0.19)
-7101.38
-26357.96
-33715.31
-13322.10
-8210.76
-6713.13
-37326.50
-8645.07
Ours
(0.07)
(0.06)
(0.10)
(1.34)
(0:23)
(0.32)
(1.39)
(0.69)
Table 4: Comparison of mean log-likelihood (MLL) and runtime between ARTree and MDTree under RWS and VIMCO optimization, each trained for 400,000 iterations. MDTree consistently achieves higher MLL and reduces runtime by over 40% compared to ARTree.
Methods
MLL
Runtime (s)
ARTree
-7107.74
128.7
MDTree__re rws
-7103.71
75.0(141.72%)
ARTree vimco
-7106.59
114.7
MD'ree__ vimco
-7101.38
63.7(144.46%)
Phy performs the worst with consistently low and unstable values. Fig. 5 in MLL, quickly reaching and maintaining high scores, whereas ARTree, SBN, and especially GeoPhy lag behind.
highlights MDTree's advantages
Table 5: Evaluation of ELBO (1) on eight datasets. Higher values indicate better performance. Results for GeoPhy were not reported in its original publication and are reproduced by us. Light gray marks the best baseline result, and darker gray marks the best overall result. Our method consistently achieves the highest ELBO across all datasets.
Dataset
DS1
DS2
DS3
DS4
DS5
DS6
DS7
DS8
Methods
#Taxa,
(27,1949)
(29,2520)
(36,1812 )
(41,1137)
(50.378)
(50.1133)
(59,1824)
(64,1008)
MCMC-
-7110.24
-26368.88
-3373.222
-13331.83
-8217.80
-6728
-37334.85
-8655.05
SBN
based
(0.03)
(0.03)
(0.02)
(0.02)
(0.04)
(0.04)
(0.03)
(0.04)
ARTree
77110.09
26368.78
33735.22
13330.27
8215.34
6725.33
-3332254
8651.73
Structure
(0.04)
(0.07)
(0.08)
(0.05)
(0.04)
(0.06)
(0.13)
(0.05)
Generation
-7166.67
26434.84
33766.72
-33389336
8220.91
6769.41
-37882.9
8654.39
GeoPhy
(1.71)
(0.10)
(0.15)
(3.45)
(2.64)
(3.25)
(1.97)
(0.97)
77055.88
-26362.75
-33430.94
-13113.03
-8053.23
-6324.90
-36838.4 42
-8409.06
Ours
(0.06)
(0.12)
(0:34)
(3.65)
(2.57)
(1.26)
(1.99)
(1.09)
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Comparison of ELBO Across Models for DS1
Comparison of MLL Across Models for DS1
-6500
-6500
-7000
-7000
Mean ELBO
Mean MLL
-8000
-8000
ARTree
ARTree
SBN
9000
9000
GeoPhy
GeoPhy
MDTree
MDTree
-10000
-10000
0.2
0.4
0.6
1.0
0.2
0.4
0.6
0.8
1.0
Steps
Steps
Figure 4: Comparison of ELBO.
Figure 5:
Comparison of MLL.
5.3 Runtime Reduction and Efficiency Evaluation (RQ2)
MDTree demonstrates substantial runtime efficiency across all datasets, outperforming ARTree consistently, Both runtime and the number nodes are log-transformed on the vertical axcs, with solid and dashed lines representing the RWS and VIMCO optimization techniques. MDTree achieves faster than ARTree across all datasets, with VIMCO providing further reductions, especially for MDTree-VIIN VIMCO, which exhibits the lowest runtime. The efficiency of MDTree becomes even more apparent dataset complexity increases. Table ! confirms this finding, with MDTree reducing runtime by 41.72% (RWS) and 44.46% (VIMCO) compared to ARTree while maintaining superior MLL metrics. This underscores MDTree's efficiency and scalability, particularly with VIMCO optimization.
5.4 Tree Parsimony in Phylogenetic Inference (RQ3-1)
To evaluate the parsimony of tree structures generated by the model, follow established methodolo- gies (Zhou et al. 2024 minimizing the genetic mutations required to infer the optimal tree. The parsimony score evaluates how well the generated tree adheres to the principle of minimizing evolutionary changes, where fewer mutations are assumed to explain the observed genetic data better. We compare the results against the most parsimonious tree identified by the traditional PAUP* tool (Soofford] 1998.. The parsimony score in Fig. denotes the minimum mulations of genetic changes needed to account for the evolutionary relationships in the data. Since scores are plotted as negative values, lower scores indicate more complex trees and, consequently, poorer model performance. MDTree and ARTree achieved higher scores (approach- ing -4000) in fewer steps, reflecting simpler and more parsimonious trees. In contrast, PhyloGFN exhibited early fluctuations and ultimately stabilized around -5000, indicating suboptimal performance compared to others.
5.5 Tree Topological Diversity in Generated Trees (RQ3-2)
To assess the diversity of tree topologies generated by MDTree, we use three metrics: Simpson's Diversity Index (He & Hu 2005), Top Frequency, and Top 95% Frequency, as detailed in Table i A higher Diversity Index, which approaches 1, suggests broad diversity among generated tree topologies. A largerr number of topologies in the Top 95% Frequency implies the generated trees more varied and distributed across many unique structures. Conversely, lower Top Frequency suggests the absence of a dominant tree structure, pointing loward a more balanced generation. For instance, in DS3, with 36 species sequences, the Top 95% Topologies metric reveals 1,146 distinct tree structures, indicating a wide range of possible phylogenetic solutions. MDTree achieves a Diversity Index close to 1, showcasing capacity for generating highly diverse topologies even in complex datasets. Furthermore, the Top Frequency metric remains notably low, further reinforcing the diversity and indicating that single tree topology is overly dominant.
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Table 6: Topological comparison of three tree diversity metrics. Higher values of Simpson's Diver- sity Index and the number of topologies accounting for the top 95% cumulative frequency indicate better diversity. In contrast, a lower frequency of the most frequent topology reflects balanced distribution.
Dataset
Statistics
MrBayes
ARTree
Ours
Diversity Index (f)
0.87
0.89
0.99
DS1
Top Frequency (1)
0.27
0.1
0.007
Top 95% Frequency (1)
42
10
121
Diversity Index (f)
0.89
0.96
0.99
DS2
Top Frequency (L)
0.27
0.43
0.13
Top 95% Frequency (f)
208
203
301
Diversity Index (f)
0.98
0.89
0.90
DS3
Top Frequency (L)
0.02
0.01
0.004
Top 95% Frequency (1)
753
509
1146
Diversity Index (f)
0.86
0.89
0.99
DS4
Top Frequency (1)
0.11
0.05
0.002
Top 95% Frequency (t)
4169
4125
8746
Comparison of Parsimony Score for DS1
Negative Parsimony Score
-4000
-4500
+302 par.
-5000
1.25x speed
+510 par.
-500
1.5x speed
ARTree
MDTree
PhyloGFN
-6000
20
40
60
80
100
Steps
Figure Comparison of negative parsimony scores on the DS1 datasct. The parsimony score
denotes the minimum number of variation steps required to interpret each tree. The lower the negative score, the poorer the model performance.
5.6 Bipartition Frequency for Tree Quality (RQ3-3)
In phylogenetin analysis, a bipartition refers to dividing taxa (species or genes) into two groups on either side of a node within the tree. When multiple tree samples are generated, as in Bayesian inference methods like MrBayes, each sample may have a different topology. Bipartition frequency quantifies how often a specific bipartition appears across all samples, providing insight into the support for particular evolutionary relationships. We use this bipartition frequency distribution to assess the model's ability to capture phy- logenetic relationships, as shown in Fig. The horizontal axis indicates the bipartition rank within the tree topology, while the vertical axis displays the normalized occurrence frequency of cach bipartition. The MDTree and MrBayes curves are closely aligned, indicating that MDTree's results closely malch those of the widely accepted gold standard. In contrast, the ARTree method shows a noticeable deviation, espe- cially the higher-ranked bipartitions, demonstrating that MDTree offers improved accuracy over ARTree in capturing evolutionary structures. This suggests that MDTree captures the evolutionary patterns with greater accuracy compared to ARTree.
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DS1
DS2
DS3
DS4
MiMaet ACTCE
Mikayes AfTher Ours
MiDlets Affice
10-
MrBarts Affiee
Norma ized Frequency
Normalized Frequency
Norma linlllizdd Frequency
Normalized Frequency
2-8
0.2
0.8
0.9
0.4
0.4
0.
20
66
100
120
140
100
10
30
66
69
10
25
30
16
60
70
25
53
108
152
Bipartition Rank
Bipartition Rank
Bipartition
Rank
Bipartition Rank
Figure 7: Bipartition frequency distribution of tree topologies. The closer the two curves are, the better.
Table 7: Comparison of different genomic language models (LMs) as structure generators in our frame- work, evaluated on Mean Log- Likelihod (MLL, f) and Evidence Lower Bound (ELBO, f). Models include DNABERT2, HyenaDNA, and NT. Higher values indicate better performance. DNABERT2 achieves the highest MLL and ELBO among the tested models, indicating its superior ability to capture genomic sequence patterns beneficial for phylogenetin inference.
Method MLL(1)
ELBO(t)
DNABERT2
-7101.38
-7005.98
HyenaDNA
-7109.36
-7014.17
NT
-7111.07
-7017.11
5.7 Analysis and Ablation (RQ4-1)
We compare MDTree with three other schemes, yiclding the following observations: (i) Removing opti- mization techniques like RWS VIMCO led to performance drop of 5.21 in MLL, as shown by slight fluctuations in the MLL curve in Fig. T highlighting their role in stabilizing convergence. (ii) Excluding the LAX model of VIMCO optimization caused decrease of 2.36 in MLL and 1.88 in ELBO, indicating its effoctiveness in reducing variance during discrete sampling. (iii) Table 7 and Table g show that the removal of the DON results in the most signíficant impact, with a drop of about 3.67 in MLL, underscoring its critical role in optimizing node addition order and improving tree generation. Overall, the full MDTree consistently achieves the best across both metrics. We select the genome-specific foundation model DNABERT2 for our phylogenetic inference research. Although models like HyenaDNA (Nguyen al. 2023 and Nucleotide Transformer (NT) (Dalla- Torre et al. 2023 excel in long-sequence modeling, they are less apt for our spe- DNABERT2 outperforms others, likely due to its specific optimization for
cific needs. As shown in Table genomic data.
5.8 Visualization of PhyloTree Structure Rerl-World Data (RQ5)
To assess the biological relevance of the tree structure generated by MDTree, we applied it construct phylogenetic tree for an Angiosperms353 genomic dataset (Zuntini et al. 2024 The tree successfully recov- ered major branches within the order Rosales, revealing distinct evolutionary lineages, including Rosaceae, Moraccae, and Polygonaccac familics. As shown in Fig. 10 the genera Polygala vulgaris and Polygala bal- duinii are clearly separated from other groups, consistent. with their classification in the Potentillaceae family. The remaining groups, distinguished by color, represent genera within the Rosaceae and Moraceae families, such as Rosa, Rubus, Ficus, and Adansonia. In Rosaccae, genera like Rosa, Rubus, and Prunus highlight their common ovolutionary ancestry, while in Moraccae, Ficus and Broussonetia reflect the internal diversity and evolutionary divergence within the family.
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Figure 8: Ablation study MDTree on four datasets, reported mean log-likelihood (MLL) and ELBO (higher is better). We evaluate the impact of removing the optimization phase, removing LAX in VIMCO, and removing the Dynamic Ordering Network (DON), The last column shows the average MLL across datascts, with green values indicating the drop compared to the full MDTree.
DS1
DS2
DS3
DS4
Method
Average
MLL
ELBO
MLL
ELBO
MLL
ELBO
MLL
ELBO
MDTree
-7101.38
-7005.98
-26357.5
-26362.75
-33155331
-33430.44
-13322.10
-13113.03
-20051.18
w/o optimization
-7106.59
-7010.34
-26377.00
-6377.01
-3373.225
33447.94
-13339.71
-13130.01
-20064.11
(-12.93)
vimco w/o Lax
-7103.74
-7007 86
-2636188
26368.52
-33718.00
33436.07
-13326.95
-13118.00
20055.22
(-4.04)
w/o DON
-7105.05
-7010.02
-23366447
26372.04
-33723.67
-3439.15
-13332.88
-13121.33
-20058.77
(-7.59)
MLL Comparison for Different Configurations
-6600
6800
Mean MLL
-7000
-7200
MDTree
-7400
MDTree w/o DON
-7600
-7800
MDTree optimization
8000
0.0
0.2
0.4
0.6
0.8
1.0
Steps
Figure Ablation of different modules. MDTree w/o optimization curve exhibits slight fluctuations, emphasizing the importance of optimization techniques in improving stability.
6 Conclusion and Limitation
In this paper, we present MDTree, a novel framework that redefines phylogenetic tree generation as Dy- namic Autoregressive Tree Generation task. By leveraging Diffusion Ordering Network to learn biologically informed node orders directly from genomic sequences, MD'rre overcomes the limitations of fixed or random node orders. integrates GNNs and Language Models to caplure complex tree lopologies, while a Dynamic Masking Mechanism enables parallel node processing, improving computational efficiency. Experiments on phylogenetic benchmarks show MDTree achieves state-of-the-art performance.
MDTree has yet to be applied to other sequence lypes, such as protein sequences. Future work will explore multimodal approaches, integrating genomic and protein data for more comprehensive evolutionary tree construction, as well as scaling the model for complex evolutionary scenarios.
Author Contributions
Zelin Zang and Chenrui Duan contributed equally to this research. Z.L., J.W., J.L., Z.Lei and S.Z.L. conceived and supervised the project. Z.Z. and C.D. designed the model and C.D. conducted code and experiments. 7.7.,, C.D., J.W., J.L,, Z.Tei and S.Z.L. wrote the manuscript with input from all authors.
Acknowledgments
This work was supported in part by National Science and Technology Major Project (No. 20222D0115101),
National Natural Science Foundation of China Project (No. U21 A20427), Project (No. WU2022A009) from the Center of Synthetic Biology and Integrated Bioengineering of Westlake University and InnoHK Program. We thank the AT Station of Westlake University for the support. of GPUs.
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Polygala volygala_vulgaris
Polygala balduinii _ialyallabbaduinni
Rosa roses
Rubus Rubus_black
blackberries
Rubus __rambleebbbb
@isaccaa_cherriee
Rosaceae_almonds
Rosáceae almonds
Rosaceae apricots
Rosaceae Prunus_avium
Rosaceae Prunus_plume
Rosaceae malus
Moraceae figs
@horaceaeFFicus _iccu_cccciiceeeee
Moraceae Morus muuubrrryr
Moraceae jackfee
Moraceae breadfruit
Moraceae heterophyllus
Grandisonia iaantisonia_altermaans alternnn
Amphiuma tridacty um
Figure 10: Visualization of generated phylogenetic tree for subset of species from the An- giosperms353 dataset. Different colors indicate distinct plant families genera, illustrating the model's ability to cluster related species into coherent subtrees. For example, species within the gemus Rubus (blue) and family Moraceae (green) are correctly grouped together, reflecting biologically plansible evolutionary relationships. This demonstrates that the proposed method can recover meaningful phylogenctic structure consistent with known laxonomy.
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20


Reviews:
[{"review_id": "19sXCg8v45", "reviewer": "Reviewer_Dumc", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes MDTree, a framework for phylogenetic inference with the goal of improving accuracy and efficiency of inferring tree topologies and branch lengths from genomic data. Given a set of species and their genomic representations, the task is to learn a phylogenetic tree capturing evolutionary relationships. The paper's main contributions are in combining generative modeling techniques with biologically informed priors. The dynamic ordering network learns node addition orders and positions, addressing the fixed-order bias of previous work with autoregressive models. The dynamic masking allows for parallel node insertions to improve efficiency. The dual-pass refines the branch lengths for precise tree generation. Empirical results show improvements in KL divergence and ELBO + MLL metrics. Specfically, MDTree outperforms the autoregressive baseline ARTree while reducing runtime by over 40%. \n\nStrengths: The paper is well-motivated, addressing the exposure bias from traditional approaches for autoregressive phylogeny generation. The experimental setup is strong, with ample comparisons to relevant baselines. The paper also presents ablations, highlighting the importance of the dynamic ordering network. The authors also validate the biological relevance of their method by recovering ground-truth relationships from the Angiosperms353 genomic dataset.\n\nWeaknesses: The evaluation on real biological data is limited, with results only on the Angiosperms353 dataset. Also, it is unclear if the method's efficacy is specific to the phylogenetic inference task or if it would also apply to other domains. Although, this is a limitation the authors point out for future work.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The experimental results consistently support the paper's claims of improved accuracy and runtime. Ablations demonstrate the importance of DON, masking, and optimization methods. The evaluation is robust, covering both likelihood and topological metrics. One downside is the scale of the data; the authors mentioned the goal to extend phylogenetic inference for high numbers of taxa but the evaluation here is only shown for up to 64 taxa.", "audience_yn": "Yes", "audience_explanation": "The paper is relevant for the intersection of machine learning and computational biology, and potentially more broadly for inference involving structured data. It may interest readers working on autoregressive generative models, diffusion-based ordering, and graph-based inference. But, one limitation of the appeal is most of the contributions being specific to phylogenetic inference.", "requested_changes": "Critical for recommending acceptance:\nStronger validation on larger or more diverse phylogenetic datasets (eg more taxa) to compare scalability.\nClarify statistical uncertainty, eg std errors for reported numbers in table 2 and 3 (since many of the differences are relatively small)\n\nWould strengthen the work:\nExpanding the biological interpretation section beyond a single dataset.\nAdding more details on training implementation and hyperparameter tuning for reproducibility.\nDiscussing the limitations of relying on pretrained genomic language models.", "broader_impact_concerns": "No concerns on ethical implications.", "additional_comments": "The paper is harder to follow for a reader unfamiliar with the task of phylogenetic inference; it would be helpful to simplify the notation and introduce the definitions earlier in the paper.", "extra_scores": {}, "extra_text": {}}, {"review_id": "UHrrnwoDRv", "reviewer": "Reviewer_srMK", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposed a new method for phylogenetic inference, which uses a dynamic ordering network to output orders of items, and use a subsequent network to assign items onto a tree. \nStrength:\nThe idea of ordering the items and then constructing a tree is interesting, and more reasonable than eg ARTree. \nThe paper shows promising experiment results on some metrics.\nWeakness:\nThe paper is not well written, many details are missing or confusing, see next sessions.", "claims_and_evidence_yn": "Yes", "claims_explanation": "According to the authors, the proposed method should be more accurate in prediction performance and faster. It can be seen that the method is faster and more accurate from figure 2, as well as table 2, 4, 6. On table 3 and 5 the results seem very marginal.", "audience_yn": "Yes", "audience_explanation": "I am not entirely sure, but I can assume that some people who work on machine learning for science will find this paper useful.", "requested_changes": "Maybe add a background section before your method part, about the fundamentals of phylogeny and machine learning for phylogeny, so that people who work on machine learning without a background in biology can also have a high level overview. \nAnd please address some questions about details of the method:\n1. ARTree adds species sequentially according to a predefined order, what order is that? Can't it be ordered heuristically by some concepts, e.g., mammals first, reptiles second, etc?\n2. The DON outputs the ordering of the species, that is fine, but is it permutation invariant for the downstream models? For example, in figure 1(b), mammals first, birds next, reptiles finally, does it make a difference if it predicts reptiles first? The ordering is not well convincing. \n3. The tree construction module allows some parallelization, but how to define how many nodes to insert? Can you parallelize it even more? \n4. I think the graph construction in figure 3 A strongly depends on the LM. Are there any reason on selecting a specific LM for that? Why not directly output orderings given the LM? You can finetune the LM so that it gives you an ordering given previous predicted species. Also, a higher level question is, why should we train a DON? Why not just use genetic inference tool e.g. MrBayes to give the ordering? \n5. The method section needs a lot of polishing in my opinion. The concepts and methodology are not well written, e.g. equation 3,4 I don't understand, and I don't know what is h_i in equation 7 even I know MHA.", "broader_impact_concerns": "Not discussed in the paper. This work is potentially useful for biology or taxonomy, accelerating science discovery.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "g2JTfdXwoL", "reviewer": "Reviewer_2j25", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "# Summary\n\nThis paper introduces MDTree, a new way to build phylogenetic trees. The idea is to fix two problems with older autoregressive (AR) models: (1) they build trees in a fixed order that may not make sense biologically, and (2) they insert one node at a time, which can be slow.\n\nMDTree has three main tricks:\n1. Dynamic Ordering Network (DON): learns a better order to add taxa, using genomic embeddings (from DNABERT2) and a diffusion process to guide the sequence.\n2. Dynamic masking + parallel insertion: instead of adding just one node per step, it can add multiple at once, making things faster.\n3. Dual-pass traversal: runs both postorder and preorder passes to estimate branch lengths, so it handles both tree shape and branch lengths together.\n\nThe model is tested on eight benchmark datasets. It tends to get better likelihood scores and lower KL divergence compared to older methods like ARTree, while also being faster. There’s also a case study on Angiosperms showing the method can recover biologically plausible groupings.\n\n# Strengths\n1. Clever tree building order. The ordering module is an interesting novel idea, instead of forcing a fixed sequence, it learns an order that’s more biologically sensible, which avoids the usual exposure bias problem in AR tree building.\n2. It handles both tree topology and branch lengths end to end, which is a nice step up from models that only do the topology part.\n3. The paper reports a wide range of metrics like KL divergence, ELBO/MLL, parsimony, runtime, diversity, and even shows a biological case study. That makes the evaluation comprehensive.\n\n# Weaknesses\n1. Key technique ordering supervision is a bit circular. The ordering network is trained with “reference orders” produced by tools like MrBayes. That’s a strong baseline itself, and it raises a question: if you already need to run MrBayes to train MDTree, are you really saving time? This could limit how practical the method is in the wild.\n2. Experiments: The paper shows higher likelihoods, but unless I missed anything, it doesn’t really validate if the branch lengths are numerically accurate on simulated data with ground truth. Also, key design choices (like the cosine mask schedule for parallel insertion) don’t get sensitivity tests, so it’s hard to know how robust those choices are.", "claims_and_evidence_yn": "Yes", "claims_explanation": "Mostly yes. The results are pretty strong against ARTree, both in accuracy and runtime. But the efficiency claim would be more convincing if runtime was reported end-to-end (including embedding time and ordering inference). The biological case study is a nice touch but feels more anecdotal than conclusive.", "audience_yn": "Yes", "audience_explanation": "Yes. People in machine learning who work on structured prediction, generative models, or biology would definitely be interested. The ideas of dynamic ordering and parallel insertion are general enough that they could matter beyond phylogenetics.", "requested_changes": "1. Be clearer about ordering supervision. (Critical)\n\nEither show results where DON is trained without external reference orders, or be upfront about the real cost of getting those orders. Otherwise it feels like MDTree depends on running an expensive baseline first, which hurts the story.\n\n2. Validate branch lengths and test sensitivity. (Strengthen the work)\n\nRun on simulated data where ground truth branch lengths are known, so you can measure absolute accuracy. Also, test how sensitive the results are to design choices like the masking schedule or bias terms. This would better support the robustness of the method.", "broader_impact_concerns": "NA.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)