FLORAH-Tree: Autoregressive Halo Merger Trees

• FLORAH-Tree is a likelihood-based autoregressive generative model that produces complete dark-matter halo merger trees capturing the hierarchical buildup of cosmic structures.
• It utilizes a GRU-based history encoder, a multiplicity classifier, and a conditional normalizing flow to accurately model progenitor properties and merger statistics.
• The model offers a fast, simulation-like alternative to expensive N-body simulations and EPS methods, ensuring statistical fidelity for semi-analytic galaxy predictions.

Searching arXiv for FLORAH-Tree and related predecessor work. FLORAH-Tree is a likelihood-based autoregressive generative model for complete dark-matter halo merger trees. It was introduced as an extension of the earlier FLORAH framework, which modeled only main progenitor branches, and it aims to replace either expensive simulation-extracted trees or less accurate semi-analytic Extended Press–Schechter (EPS) trees with fast samples that retain the statistical fidelity of $N$-body merger histories, including dependence on secondary halo properties. Trained on merger trees from the Very Small MultiDark Planck (VSMDPL) cosmological $N$-body simulation, FLORAH-Tree is reported to reproduce merger statistics and downstream semi-analytic galaxy predictions substantially better than an EPS baseline (Nguyen et al., 14 Jul 2025, Nguyen et al., 2023).

1. Definition, scope, and disambiguation

FLORAH-Tree addresses the problem of generating complete halo merger trees, not merely one-dimensional assembly histories. In this context, a merger tree records the hierarchical buildup of a halo across cosmic time: a halo at low redshift has one or more progenitors at earlier redshift, each of which in turn has its own progenitors. Such trees are essential inputs to semi-analytic models (SAMs) of galaxy formation. The stated motivation is that simulation-extracted trees are accurate but expensive and limited in mass-volume-time dynamic range, whereas EPS-based Monte Carlo trees are cheap but rely on ad hoc assumptions, typically Markovianity, and generally miss environmental or secondary-property correlations while mispredicting merger statistics (Nguyen et al., 14 Jul 2025).

The model extends the original FLORAH of Nguyen et al. from main progenitor branch generation to full branching trees. The predecessor framework was explicitly presented as a first step toward a future system for “planting full merger trees,” and FLORAH-Tree realizes that next step by adding a classifier for progenitor multiplicity, an autoregressive progenitor-property generator for multiple progenitors, and conditioning on the descendant’s assembly history rather than only on the current halo (Nguyen et al., 2023).

Despite the botanical resonance of its name, FLORAH-Tree is unrelated to plant-identification or forest-remote-sensing systems such as “Floralens” (Filgueiras et al., 2024) or “FLORA” (Vautier et al., 30 Jun 2026). In the arXiv literature, the term denotes a cosmological graph or tree generator specialized to dark-matter halo assembly (Nguyen et al., 14 Jul 2025).

2. Tree representation and probabilistic formulation

Although motivated as a graph generative model, FLORAH-Tree represents merger histories as rooted directed trees evolving backward in time. Each node is a distinct dark-matter halo; subhalos are removed, so the paper works with isotrees, where once a subhalo merges into a distinct halo, it is no longer tracked. Each node has feature vector

$\mathbf{x} = (M_{\rm vir}, c_{\rm vir}),$

where $M_{\rm vir}$ is virial mass and $c_{\rm vir} \equiv R_{\rm vir}/r_s$ is NFW concentration. The choice of concentration is inherited from the original FLORAH work, where it was found necessary to capture assembly bias (Nguyen et al., 14 Jul 2025).

Edges connect each descendant halo at lower redshift to its progenitors at higher redshift, and both training and sampling proceed backward in time. The temporal axis is discretized on simulation snapshots, with time encoded in practice by the scale factor $a=1/(1+z)$, since it is more linear with histories. If snapshots are removed by subsampling, progenitors from retained snapshots are directly linked to the next retained descendant to preserve causal ancestry (Nguyen et al., 14 Jul 2025).

A central design choice is the mass ordering of progenitors. For a descendant halo, progenitors are sorted by descending mass,

$M_1 > M_2 > \cdots > M_{N_p},$

which turns an unordered progenitor set into an autoregressive sequence initialized by a dummy start token $\mathbf{x}_0=\vec{0}$. The conditioning context for a given descendant is its unique branch history from the root to that descendant,

$$\mathbf{X}_{\rm hist} = \{\mathbf{x}^{(1)}, \mathbf{x}^{(2)}, \dots, \mathbf{x}^{(N_{\rm hist})}\},$$

together with the corresponding redshift vector and the target progenitor redshift $z_p$. The model conditions only on that branch history; other branches are not input when predicting the node’s progenitors (Nguyen et al., 14 Jul 2025).

The paper factorizes the conditional distribution of progenitors and progenitor count as

$N$0

with an autoregressive decomposition of progenitor properties,

$N$1

This formulation makes tree generation a recursive “sample multiplicity, then sample children in order” process. Environment is included only indirectly through $N$2; no direct large-scale density field or initial density field is used in the present model (Nguyen et al., 14 Jul 2025).

3. Architecture and training pipeline

FLORAH-Tree has three jointly trained components: a history encoder $N$3, a multiplicity classifier $N$4, and a progenitor-property neural density estimator consisting of a progenitor-sequence encoder $N$5 together with a conditional normalizing flow $N$6. The history encoder maps the branch history and times to a latent context vector,

$N$7

Its architecture is a 4-layer GRU with 128 hidden units per layer and ReLU activations; halo features and redshift or time are each projected by fully connected layers and then summed before entering the GRU, and only the final timestep output is retained (Nguyen et al., 14 Jul 2025).

The multiplicity classifier models $N$8 as a categorical distribution over progenitor counts $N$9, with $\mathbf{x} = (M_{\rm vir}, c_{\rm vir}),$0 in this work. Progenitor redshift $\mathbf{x} = (M_{\rm vir}, c_{\rm vir}),$1 is projected to 128 dimensions and added to the history embedding. The classifier itself uses four fully connected layers with hidden size 16 and GELU activation (Nguyen et al., 14 Jul 2025).

The progenitor-property model uses a second GRU-based encoder, architecturally matched to the history encoder, to summarize previously generated progenitors together with $\mathbf{x} = (M_{\rm vir}, c_{\rm vir}),$2 and a one-hot encoding of $\mathbf{x} = (M_{\rm vir}, c_{\rm vir}),$3. Conditioned on that state and on the history embedding, the model represents the density of the $\mathbf{x} = (M_{\rm vir}, c_{\rm vir}),$4-th progenitor through a conditional normalizing flow. The flow uses four neural spline flow layers with monotonic rational-quadratic spline transforms, eight knots per spline, and hidden size 128. The paper emphasizes that conditioning the flow on $\mathbf{x} = (M_{\rm vir}, c_{\rm vir}),$5 is crucial because effective mass-partition constraints change the plausible progenitor mass distribution depending on whether a descendant splits into one, two, or three progenitors (Nguyen et al., 14 Jul 2025).

The authors also experimented with a Transformer variant in which the history encoder acted as encoder and the progenitor encoder as decoder, and they also tried sinusoidal redshift encoding. Performance was similar, but GRUs were faster, so the GRU version was adopted. The total trainable parameter count is $\mathbf{x} = (M_{\rm vir}, c_{\rm vir}),$6, divided into $\mathbf{x} = (M_{\rm vir}, c_{\rm vir}),$7 for the history encoder, $\mathbf{x} = (M_{\rm vir}, c_{\rm vir}),$8 for the neural density estimator, and $\mathbf{x} = (M_{\rm vir}, c_{\rm vir}),$9 for the classifier (Nguyen et al., 14 Jul 2025).

Training data come from the Very Small MultiDark Planck simulation, a dark-matter-only $M_{\rm vir}$0-body run with $M_{\rm vir}$1 particles, particle mass $M_{\rm vir}$2, box $M_{\rm vir}$3, and Planck-like cosmology

$M_{\rm vir}$4

Halos are identified with Rockstar and trees with Consistent-Trees. Preprocessing removes all subhalos, keeps only distinct-halo trees, discards trees whose root has fewer than 500 particles, discards any halo with fewer than 200 particles and all its progenitors, and removes halos with poorly fit concentrations $M_{\rm vir}$5 together with all their progenitors. The authors explicitly do not correct tree-level numerical artifacts such as mass loss or fragmentation in the training set (Nguyen et al., 14 Jul 2025).

The training and validation set consists of approximately $M_{\rm vir}$6 trees from a $M_{\rm vir}$7 sub-volume, split 80/20, while the test set consists of approximately $M_{\rm vir}$8 trees from a separate non-overlapping $M_{\rm vir}$9 sub-volume. To reduce cost and avoid overfitting to exact snapshot spacing, training randomly subsamples each tree at intervals of 2–6 snapshots up to $c_{\rm vir} \equiv R_{\rm vir}/r_s$0, whereas testing uses every 4 snapshots to $c_{\rm vir} \equiv R_{\rm vir}/r_s$1, giving 27 snapshots (Nguyen et al., 14 Jul 2025).

The classifier is trained with cross-entropy,

$c_{\rm vir} \equiv R_{\rm vir}/r_s$2

and the neural density estimator with teacher-forced negative log-likelihood,

$c_{\rm vir} \equiv R_{\rm vir}/r_s$3

The total loss is

$c_{\rm vir} \equiv R_{\rm vir}/r_s$4

Optimization uses AdamW with peak learning rate $c_{\rm vir} \equiv R_{\rm vir}/r_s$5, weight decay $c_{\rm vir} \equiv R_{\rm vir}/r_s$6, cosine annealing, 25,000 warm-up steps, 500,000 decay steps, and batch size 128. Training converges in about 72 hours on one NVIDIA Tesla A100 GPU (Nguyen et al., 14 Jul 2025).

4. Generation procedure and evaluation methodology

Given root halos and a sequence of redshifts, FLORAH-Tree generates a merger tree snapshot by snapshot backward in time. For each descendant halo, the model encodes its branch history, samples the number of progenitors from the classifier, and then generates progenitor features autoregressively from the conditional flow starting with $c_{\rm vir} \equiv R_{\rm vir}/r_s$7. Descendant–progenitor links are recorded, and the process is repeated recursively for all newly generated progenitors at the next earlier snapshot. A branch stops when sampled mass falls below the minimum threshold $c_{\rm vir} \equiv R_{\rm vir}/r_s$8. If sampled progenitors violate mass ordering, they are resampled; the paper states that this is rare (Nguyen et al., 14 Jul 2025).

The main comparison baselines are held-out simulation trees from VSMDPL and EPS trees generated with the Parkinson et al. (2008) algorithm as implemented in SatGen. For fair comparison with EPS, the setup matches the same root halo mass distribution, the same redshift range, the same minimum halo mass $c_{\rm vir} \equiv R_{\rm vir}/r_s$9, and the same progenitor cap $a=1/(1+z)$0 (Nguyen et al., 14 Jul 2025).

Evaluation is intentionally astrophysically targeted rather than graph-theoretically generic. The reported diagnostics are progenitor-descendant mass ratio distributions, progenitor-progenitor mass ratio distributions, volumetric merger rates, and downstream Santa Cruz semi-analytic model scaling relations. No separate likelihood-based calibration metrics or graph-distance metrics are used (Nguyen et al., 14 Jul 2025).

For progenitor statistics, the paper defines

$a=1/(1+z)$1

with $a=1/(1+z)$2 for $a=1/(1+z)$3. Merger rates are studied via the progenitor-to-primary mass ratio

$a=1/(1+z)$4

and the volumetric merger rate $a=1/(1+z)$5, formed by summing the contributions from $a=1/(1+z)$6 and $a=1/(1+z)$7 progenitors. For downstream physics, the SAM is run using only the halo properties available for all methods, namely $a=1/(1+z)$8 and $a=1/(1+z)$9 (Nguyen et al., 14 Jul 2025).

5. Empirical results and astrophysical significance

FLORAH-Tree is reported to reproduce the distributions of $M_1 > M_2 > \cdots > M_{N_p},$0 and $M_1 > M_2 > \cdots > M_{N_p},$1 well across the full tested mass and redshift range. The main discrepancy appears in the tail of $M_1 > M_2 > \cdots > M_{N_p},$2, especially $M_1 > M_2 > \cdots > M_{N_p},$3, corresponding to apparent descendant mass loss relative to the main progenitor; the authors attribute this primarily to simulation or tree-construction artifacts rather than to a modeling failure. Across the test set, halo mass loss occurs about 17% of the time for $M_1 > M_2 > \cdots > M_{N_p},$4 primary progenitors and 0.04% for $M_1 > M_2 > \cdots > M_{N_p},$5 secondary progenitors. The paper also notes that performance remains good even though only about 0.7% of training trees have $M_1 > M_2 > \cdots > M_{N_p},$6 (Nguyen et al., 14 Jul 2025).

The merger-rate results are one of the model’s central claims. FLORAH-Tree merger rates agree closely with VSMDPL across descendant mass, merger ratio, and redshift slices, whereas EPS/P08 trees systematically overpredict merger rates. The paper states this high-redshift EPS bias inconsistently—once as as much as $M_1 > M_2 > \cdots > M_{N_p},$7 dex in a figure caption and once as up to 50% in the main text—but both statements indicate a clear high-$M_1 > M_2 > \cdots > M_{N_p},$8 excess in the EPS baseline. A shared caveat for both simulation and model is incompleteness of minor mergers near the 200-particle threshold, which causes the low-$M_1 > M_2 > \cdots > M_{N_p},$9 merger rate to flatten or decrease artificially, especially at high redshift and low mass (Nguyen et al., 14 Jul 2025).

A stronger validation comes from coupling the generated trees to the Santa Cruz semi-analytic model. FLORAH-Tree + SC-SAM reproduces the VSMDPL-tree SAM predictions extremely well for the median stellar-to-halo mass relation, the scatter in that relation, the correlation of stellar-to-halo-mass residual with concentration, and the mean and scatter of the supermassive-black-hole-mass–halo-mass relation. By contrast, EPS + SC-SAM recovers the stellar-to-halo mass relation only moderately well, shows noticeable low-mass discrepancies, fails badly on the residual–concentration relation, and systematically overpredicts supermassive black hole masses, which the paper interprets as consistent with its excessive merger rate (Nguyen et al., 14 Jul 2025).

These downstream results are scientifically significant because SC-SAM uses full merger histories for processes that depend explicitly on mergers, including SMBH seeding and growth, merger-triggered bright-mode black-hole accretion, merger-driven starbursts, and satellite evolution or disruption effects. The reported agreement therefore indicates not only local fidelity in halo-space statistics but also preservation of the merger information needed to reproduce galaxy–halo scaling relations in a downstream physical model (Nguyen et al., 14 Jul 2025).

In computational terms, inference for 50,000 trees takes about 16 minutes on one GPU, of which about 6 minutes is tree reconstruction. A plausible implication is that the model occupies a regime where repeated SAM studies become practical without rerunning large $\mathbf{x}_0=\vec{0}$0-body simulations. The intended use cases named in the paper include fast SAM inputs with simulation-like merger fidelity, mock lightcone construction for JWST- and Roman-era survey forecasting, interpolation across complementary simulation suites, future cosmology-conditioned emulators using CAMELS or DREAMS, and potential coupling to initial density fields via 3D CNNs for environment-consistent tree generation in specific cosmic volumes (Nguyen et al., 14 Jul 2025).

6. Limitations, assumptions, and prospective extensions

The paper is explicit about several present limitations. Temporal resolution is reduced by design: training uses random subsampling at intervals of 2–6 snapshots and testing uses every 4 snapshots, which may matter for applications sensitive to merger timing. The maximum progenitor count is hard-capped at $\mathbf{x}_0=\vec{0}$1. Only distinct halos are modeled, because subhalos are removed entirely. Environmental conditioning is indirect, entering only through $\mathbf{x}_0=\vec{0}$2; there is no explicit large-scale density field or local tidal environment. Because the training trees contain uncorrected Rockstar/Consistent-Trees artifacts, the model inherits them, with the clearest symptom being imperfect treatment of the $\mathbf{x}_0=\vec{0}$3 tail. As in any autoregressive model, error propagation across later generations is a concern, although the paper states that catastrophic accumulation is not evident in the reported statistics. Finally, the model does not explicitly enforce mass conservation via a constrained sampler, and the deployed system is trained on one simulation and one cosmology, so generalization across cosmologies is discussed as future work rather than demonstrated (Nguyen et al., 14 Jul 2025).

The relationship to the original FLORAH remains conceptually important. FLORAH generated only main progenitor branches and already demonstrated that a GRU plus conditional normalizing flow could recover mass and concentration histories, assembly-bias-sensitive clustering, and SC-SAM trends for main-branch-only inputs. FLORAH-Tree generalizes that program from sequence generation to full branching-tree generation while retaining branch-history conditioning and probabilistic sampling. In that sense, FLORAH-Tree is both a continuation of and a structural departure from its predecessor: the main-branch engine becomes one component inside a recursive rooted-tree generator (Nguyen et al., 2023).

The authors identify several directions for scaling and extension: distributed multi-GPU training, mixed precision, gradient accumulation, finer time resolution, and combinations of models across simulations of different box size and resolution. They also note that with finer snapshot spacing, events with $\mathbf{x}_0=\vec{0}$4 may become rare, potentially reducing model complexity. More ambitious future directions include cosmology-conditioned emulators trained on suites such as CAMELS or DREAMS and explicit coupling to initial density fields. This suggests that FLORAH-Tree is best understood not as a finished general theory of merger trees, but as a simulation-trained, astrophysically validated emulator of complete dark-matter halo merger trees whose current scope is already sufficient for merger-sensitive semi-analytic galaxy modeling (Nguyen et al., 14 Jul 2025).