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License Attribution 4.0 International (CC BY 4.0).

CLEF 2024: Conference and Labs of the Evaluation Forum, September 09-12, 2024,
Grenoble, France

[ orcid=0009-0002-3325-3357<EMAIL_ADDRESS>] [1]

[1]Corresponding author.

# Biomedical Nested NER with Large Language Model and UMLS Heuristics

DS@GT CLEF2024 BioNNE Competition Working Note

Wenxin Zhou  Georgia Institute of Technology, North Ave NW, Atlanta, GA 30332,
United States

(2024)

###### Abstract

In this paper, we present our system for the BioNNE English track, which aims
to extract 8 types of biomedical nested named entities from biomedical text.
We use a large language model (Mixtral 8x7B instruct) and ScispaCy NER model
to identify entities in an article and build custom heuristics based on
unified medical language system (UMLS) semantic types to categorize the
entities. We discuss the results and limitations of our system and propose
future improvements. Our system achieved an F1 score of 0.39 on the BioNNE
validation set and 0.348 on the test set.

###### keywords:

large language model prompt engineering named entity recognition nested named
entity recognition

##  1 Introduction

Named entity recognition is an information extraction task that aims to
identify named entities in text and classify them into predefined categories.
Nested named entity recognition involves detecting both the outer entities and
the inner entities. The BioNNE competition [1], being part of CLEF 2024 BioASQ
lab [2], focuses on extracting nested named entities from biomedical text. The
nested named entity types include anatomy (ANATOMY), chemicals (CHEM),
diseases (DISO), physiology (PHYS), scientific findings (FINDING), injury or
poisoning damages (INJURY_POISONING), lab procedures (LABPROC), and medical
devices (DEVICE) [3].

The challenge offers Russian, English and bilingual tracks. For the English
track, the organizers provided a training set with 50 records and a validation
set with 50 records. Each record contains a text, which is a PubMed abstract,
and a list of entities annotated in BRAT format with the starting and ending
locations of the entities in the text. In the testing phase, the organizers
released a test set with 154 abstracts and 346 extra files, which resulted in
500 records in total.

Our team focuses on the BioNNE English track. Our system uses a large language
model (specifically Mixtral 8x7B instruct model [4]) and a biomedical NER
model to find entities in an article. Then, the system uses unified medical
language system (UMLS) semantic types to filter and aggregate the entities.
The implementation can be found on Github111https://github.com/dsgt-kaggle-
clef/bioasq-2024.

##  2 Related Work

###  2.1 Nested NER

The state-of-the-art nested NER models are the machine reading comprehension
(MRC) model [5] and the sequence learning model [6]. Loukachevitch et al.
trained the MRC and sequence model on the NEREL-BIO dataset [3], which is the
predecessor of the BioNNE dataset. The two datasets are based on PubMed
abstracts and with the nested NER annotations. The difference is that the
NEREL-BIO dataset contains more entity types, including the BioNNE entity
types and additional entity types such as food, gene and activity.
Loukachevitch et al. were able to achieve a macro-F1 score of 0.5968 with MRC
model on the NEREL-BIO dataset.

###  2.2 Large Language Model

Recently, large language models (LLM) have shown great potential in solving
NLP tasks. Since they are pre-trained on a large corpus, they can solve
various problems in different domains including biomedical question answering
and information extraction. Prompt engineering [7] is one of the key
techniques to interact with LLM. The model is given a prompt with instructions
and examples. The model can then generate the response according to the
prompt. This is also known as LLM few-shot in-context learning. The quality of
the prompt affects model output and performance. As a result, a lot of
research has been done on prompt engineering to optimize the performance of
LLM on specific tasks, as an alternative to model finetuning, because
finetuning is more expensive and time-consuming.

Chen et al. [8] measured the performance of GPT-3.5 on the Biomedical Language
Understanding and Reasoning Benchmark (BLURB). GPT-3.5 achieved 58.5 while the
SOTA model had a score of 84.5. Although the existing LLM models are not as
good as the SOTA biomedical models, LLMs show great potential in text
reasoning and generation. At the same time, LLMs have limitations such as
hallucination and inconsistency [9].

Mixtral 8x7B [4], a sparse mixture of experts model (SMoE) is one of the well-
known LLM models, which can process five languages including English, French,
Italian, German and Spanish with a context of 32k tokens. The basic model and
the instruct model of Mixtral 8x7B achieve better performance than the
counterparts of GPT-3.5 [10] and Llama2 70B [11] in several benchmarks.
Mixtral 8x7B only uses 13B parameters during inference time, which makes it
more efficient than other models with the same performance. Therefore, we use
Mixtral 8x7B instruct as our LLM model in this study.

##  3 Methodology

We build a system for the BioNNE English track that uses a general-purpose LLM
and a biomedical domain language model to identify entities in an article and
then uses custom heuristics based on unified medical language system (UMLS)
semantic types to determine the entity types. The system design is shown in
Figure 1. Figure 2 shows an example of how DISO and LABPROC entities are
extracted from an article by LLM and categorized by UMLS heuristics.

Figure 1: BioNNE System Design Figure 2: BioNNE Data Flow

###  3.1 LLM and Prompt Engineering

Given an article abstract, we use Mixtral 8x7B instruct LLM model to find the
entities for each category. We construct a prompt for each category using two
of the examples in the training dataset and an instruction that describes the
entity type. Then the LLM returns a list of entities separated by semicolon in
the response. We then parse the returned entities and store them as candidates
for the category. We perform post-processing on the response list to remove
duplicates as well as entities that are not present in the original text. If
the entities in the returned string are not separated by semicolons, we
resample the LLM output. For example, Figure 3 shows the prompt and response
to query ANATOMY entity for record 26271422_en.

Prompt:

Instruction: Return phrases or entities that comprise organs, body part, cells
and cell components, body substances in TEXT, in the ANATOMY concatenated by
";" [TEXT]: Impact of bosentan therapy on stress-induced pulmonary
hypertension in patients with systemic sclerosis. AIM To describe hemodynamic
and clinical changes in patients with elevated mean pulmonary artery pressure
(MPAP) … [ANATOMY]: pulmonary; artery; pulmonary artery; lung;heart; left
heart; atrial; right atrial; cardiac; arterial; vascular; pulmonary arterial;
pulmonary vascular ### [TEXT]: The authors present the material of their study
of the morphological and molecular biological features of damage to the stem
cell … [ANATOMY]: lung biopsies; respiratory acinus; lung tissue; mesenchymal
cell; myofibroblast; mesenchymal stem cell; SCN; stem cell; cell;
lung;pulmonary; acinus; stem cell niches; tissue;mesenchymal; SCN areas;
respiratory acini; biopsies; sections; acini; cells ### [TEXT]: Endoprosthetic
replacement with lifting of abdominal wall in treatment of umbilical and
postoperative ventral hernias. The results of … [ANATOMY]:

Response:

abdominal wall defect; abdomen ptosis; polypropylene prosthesis; main flap;
roundish edges; additional flap; super lightweight; hypogastric area;
musculoaponeurotic tissues lifting; physical health component; psychic health
component; standard; original

Figure 3: Example Prompt and Response for ANATOMY entity

###  3.2 Biomedical NER model

In addition to LLM, we also use the ScispaCy biomedical NER model
en_ner_bc5cdr_md [12] to find the named entities. The ScispaCy model is
trained on the BC5CDR corpus, which detects DISEASE and CHEMICAL entities in
the document. We store the entities found by the ScispaCy model as candidates
for the DISO and CHEM categories.

###  3.3 UMLS Heuristics

Finally, we query UMLS [13] to find the semantic types for the candidate
entities. The UMLS release version used in this study is 2023AB. Specifically,
we use UMLS [14] "/search/current" API and set the search term as our entity
to retrieve the top 5 Concept Unique Identifiers (CUI) and their names
associated with the entity. For each CUI, we use the
"/content/current/CUI/cui" API to retrieve the semantic type name for the CUI
and then use the semantic_uri to find the treeid of the semantic type.
Finally, the treeid is mapped to the BioNNE category to finalize the category
of an entity.

An entity is finalized as a specific category only when both the language
model (either LLM or NER model) and the UMLS heuristics agree on the category.
If UMLS heuristics show that the entity fits multiple categories in the top 5
CUI results, we will use the category type that comes first as the final
category. The mapping between BioNNE category and UMLS semantic type is shown
in Table 1. The mapping is based on the NEREL-BIO paper [3], with additional
UMLS semantic types added for ANATOMY and LABPROC categories.

Table 1: BioNNE type and UMLS type mapping BioNNE Category | UMLS semantic types
---|---
DISO | B2.2.1.2 Pathologic Function
CHEM | A1.4.1 Chemical
ANATOMY |  A1.2 Anatomical Structure A2.1.5.2 Body Location or Region
LABPROC |  B1.3.1.1 Laboratory Procedure B1.3.1.2 Diagnostic Procedure
INJURY_POISONING | B2.3 Injury or Poisoning
DEVICE | A1.3.1 Medical Device
PHYS | B2.2.1.1 Physiologic Function
FINDING | A2.2 Finding

###  3.4 Acronym Detection

We use the ScispaCy abbreviation detector [12] to detect acronyms in the
abstract. If the long form of the acronym is identified by the language model
and UMLS heuristics as a named entity, the acronym will be assigned the same
category as the long form entity.

##  4 Results and Discussion

Our model achieved 0.348 F1 score on the leaderboard for the BioNNE English
track test set. The F1 score we achieved on the validation set is 0.39, which
is close to the test F1 score. Since the golden answers for the test set have
been not released, we will discuss the validation set results here.

The English validation set contains 50 records. The F1 scores for each
category sorted in descending order are shown in Table 2. Our model performs
well on DISO, CHEM, and ANATOMY entities, with F1 scores above 0.5, but the F1
scores for FINDING and PHYS are below 0.3.

Table 2: F1 Score for each category on the validation set Category | Precision | Recall | F1 Score
---|---|---|---
DISO | 0.7565 | 0.5613 | 0.6444
CHEM | 0.7857 | 0.4695 | 0.5878
ANATOMY | 0.8101 | 0.4082 | 0.5429
LABPROC | 0.5357 | 0.2632 | 0.3529
INJURY_POISONING | 0.2778 | 0.4545 | 0.3448
DEVICE | 0.6250 | 0.2 | 0.3030
PHYS | 0.4875 | 0.1444 | 0.2229
FINDING | 0.1288 | 0.2537 | 0.1709

We examine the model predictions for FINDING and PHYS entities and have the
following observations.

  1. 1.

The LLM model generates a lot of false positives for the FINDING entities. The
UMLS heuristics can filter out some of the false positives. However, since our
UMLS concept search does not search for exact terms, some of the LLM-
recognized phrases that are part of some FINDING entity word are not filtered
out, even though those LLM-recognized phrases (when looked at independently)
do not belong to the FINDING type. For example, for the sentence (in record
27029443_en), "The authors suggest the algorithm for choosing the order of
priority of surgical interventions on coronary and brachiocephalic arteries",
the LLM recognizes "suggest" as a FINDING entity, but the UMLS heuristics do
not filter it out because it corresponds to "Abnormal/suggest Ca", which is a
FINDING concept.

  2. 2.

PHYS entity recognition has a low recall. On one hand, PHYS entities sometimes
contain generic one-word terms such as "healthy", "lifetime", "size", "shape"
and "adults", which are not recognized by LLM or UMLS as PHYS entities. On the
other hand, some long phrases (which are outer nested entities) recognized by
LLM as PHYS entities are excluded by UMLS heuristics because the UMLS search
cannot find such terms. For example, "spirometric indicators" and "peripheral
blood oxygen saturation" in record 27030325_en are excluded by UMLS
heuristics.

The general takeaways from the results are:

  1. 1.

Context is important in NER task. The same phrases have different entity types
in different documents. For example, the words “inflammatory” and
“albuminuria” can be PHYS or DISO in different contexts. Our UMLS heuristics
only check the semantic meaning of phrases without looking at the context,
which limits the performance of our model.

  2. 2.

Our model is often unable to detect the outer nested named entities. For
example, it recognizes “cardiac contractility” as PHYS, but fails to recognize
“low cardiac contractility” as FINDING. Sometimes those outer nested named
entities are identified by LLM but rejected by UMLS heuristics because UMLS
search cannot find matching terms in the UMLS system.

###  4.1 Impact of UMLS Heuristics

To understand the impact of UMLS heuristics on the system performance, we
remove the UMLS heuristics from the system pipeline. When the LLM model
identifies multiple entity types for a phrase, we assign the final entity type
according to the INJURY_POISONING, ANATOMY, PHYS, DISO, CHEM, LABPROC, DEVICE
and FINDING order. The macro-F1 score on the validation set without UMLS
heuristics is 0.2151, which is significantly lower than the F1 score (0.348)
when UMLS heuristics are used. The F1 scores for every category are shown in
Table 3. We observe that without UMLS heuristics, the precision score for each
category becomes lower. This is because the LLM model generates many false
positives. In terms of recall, only the INJURY_POISONING and ANATOMY types
have better recall scores when UMLS heuristics are not used. This is because
we prioritize assigning the final entity type as INJURY_POISONING and ANATOMY
when the LLM detects multiple matching entity types. This result shows that
UMLS heuristics play an important role in eliminating the false positives of
LLM predictions.

Table 3: F1 Score for each category on the validation set when UMLS heuristics are not used Category | Precision | Recall | F1 Score
---|---|---|---
DISO | 0.4551 | 0.4676 | 0.4613
CHEM | 0.3731 | 0.3049 | 0.3356
ANATOMY | 0.2036 | 0.4629 | 0.2828
LABPROC | 0.1964 | 0.0973 | 0.1302
INJURY_POISONING | 0.1304 | 0.5455 | 0.2105
DEVICE | 0.069 | 0.08 | 0.0741
PHYS | 0.0742 | 0.10 | 0.0852
FINDING | 0.1357 | 0.1463 | 0.1408

###  4.2 Training Dataset and Results

The Mixtral 8x7B model officially supports five languages, which do not
include Russian. Therefore, we only attempt the English track. It is possible
that other LLM models that support Russian can be used to accomplish the
Russian/bilingual track using the same modeling pipeline. The method we use in
this study is few-shot prompt engineering and we only use two examples from
the training dataset to construct the LLM prompt. We did not include more
examples in the prompt, as the input token size of LLM is limited and the cost
of processing a long prompt is high. We also do not expect adding more
examples in the prompt would improve the performance significantly, as the
prompt is mostly used to guide the model to generate outputs in the desired
format. The entities recognized by the model are mostly based on the original
knowledge of the model. However, we believe the remaining training examples
will be useful if we want to finetune the LLM model on the BioNNE task in the
future.

##  5 Future Work

There are two directions for future improvement.

First, we can improve the heuristics for determining the category of an
entity. Our current algorithm is weak at detecting the outer nested entities.
Even though the LLM model recognizes them, we fail to assign the correct
category for those entities using the existing UMLS heuristics. We can add a
new set of heuristics for detecting the outer nested entities. Specifically,
if a long phrase contains an inner entity that is a named entity, we can
assign the outer entity with the same category as the inner entity or assign
the outer entity with a category detected by the LLM.

Second, we can improve the performance of the LLM model. Currently LLM model
generates many false positives and it may assign several entity types to a
single phrase. For example, it recognizes "neuronal dysfunction" as DISO,
ANATOMY, PHYS, FINDING and CHEM in the sentence "the severity of coronary
artery lesions and low cardiac contractility affect the degree of cerebral
ischemia and neuronal dysfunction detected by spectral EEG power" (in record
25726786_en). This is probably because the general-purpose LLM model (Mixtral
8x7B instruct) trained on the general text does not have the domain knowledge
of the biomedical field. Therefore it cannot distinguish the subtle
differences among the BioNNE entity types. We can fine-tune the LLM model on
the BioNNE training data as well as other biomedical NER datasets to improve
the performance of the model in this task.

Although the Mixtral 8x7B model does not officially support the Russian
language, a recent study by Fenogenova et al. [15] shows that Mistral 7B (a
smaller version of Mixtral 8x7B) performs well on Russian language benchmarks
compared to other LLMs. Future work can explore using Mixtral or other LLM
models to solve the BioNNE bilingual NER problems using prompt engineering
techniques in the paper or fine-tuning the model with bilingual datasets.

##  6 Conclusions

We build a system that uses a general-purpose LLM, a biomedical domain NER
model and UMLS-based heuristics to extract nested named entities from
biomedical text. Our model achieved an F1 score of 0.39 and 0.348 on the
BioNNE English validation and test sets. Although the results are not
comparable to those of the state-of-the-art models [3], that were trained on
the domain-specific dataset, our study demonstrates the potential of using
general-purpose LLM and prompt engineering with domain-specific rules to solve
biomedical NER tasks.

## Acknowledgements

We want to thank the Data Science @ Georgia Tech (DS@GT) CLEF team for their
support and acknowledge the use of Grammarly [16] to proofread this paper.

## References

  * Davydova et al. [2024] V. Davydova, N. Loukachevitch, E. Tutubalina,  Overview of bionne task on biomedical nested named entity recognition at bioasq 2024,  2024\.
  * Nentidis et al. [2024] A. Nentidis, G. Katsimpras, A. Krithara, S. Lima-López, E. Farré-Maduell, M. Krallinger, N. Loukachevitch, V. Davydova, E. Tutubalina, G. Paliouras,  Overview of BioASQ 2024: The twelfth BioASQ challenge on Large-Scale Biomedical Semantic Indexing and Question Answering,  in: L. Goeuriot, P. Mulhem, G. Quénot, D. Schwab, L. Soulier, G. Maria Di Nunzio, P. Galuščáková, A. García Seco de Herrera, G. Faggioli, N. Ferro (Eds.), Experimental IR Meets Multilinguality, Multimodality, and Interaction. Proceedings of the Fifteenth International Conference of the CLEF Association (CLEF 2024), 2024.
  * Loukachevitch et al. [2023] N. Loukachevitch, S. Manandhar, E. Baral, I. Rozhkov, P. Braslavski, V. Ivanov, T. Batura, E. Tutubalina,  NEREL-BIO: A Dataset of Biomedical Abstracts Annotated with Nested Named Entities,  Bioinformatics (2023). URL: https://doi.org/10.1093/bioinformatics/btad161. doi:10.1093/bioinformatics/btad161, btad161.
  * Jiang et al. [2024] A. Q. Jiang, A. Sablayrolles, A. Roux, A. Mensch, B. Savary, C. Bamford, D. S. Chaplot, D. de Las Casas, E. B. Hanna, F. Bressand, G. Lengyel, G. Bour, G. Lample, L. R. Lavaud, L. Saulnier, M.-A. Lachaux, P. Stock, S. Subramanian, S. Yang, S. Antoniak, T. L. Scao, T. Gervet, T. Lavril, T. Wang, T. Lacroix, W. E. Sayed,  Mixtral of experts,  ArXiv abs/2401.04088 (2024). URL: https://api.semanticscholar.org/CorpusID:266844877.
  * Li et al. [2020] X. Li, J. Feng, Y. Meng, Q. Han, F. Wu, J. Li,  A unified MRC framework for named entity recognition,  in: D. Jurafsky, J. Chai, N. Schluter, J. Tetreault (Eds.), Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics, Association for Computational Linguistics, Online, 2020, pp. 5849–5859. URL: https://aclanthology.org/2020.acl-main.519. doi:10.18653/v1/2020.acl-main.519.
  * Shibuya and Hovy [2020] T. Shibuya, E. Hovy,  Nested named entity recognition via second-best sequence learning and decoding,  Transactions of the Association for Computational Linguistics 8 (2020) 605–620. URL: https://aclanthology.org/2020.tacl-1.39. doi:10.1162/tacl_a_00334.
  * White et al. [2023] J. White, Q. Fu, S. Hays, M. Sandborn, C. Olea, H. Gilbert, A. Elnashar, J. Spencer-Smith, D. Schmidt, A prompt pattern catalog to enhance prompt engineering with chatgpt, 2023. doi:10.48550/arXiv.2302.11382.
  * Chen et al. [2023] Q. Chen, H. Sun, H. Liu, Y. Jiang, T. Ran, X. Jin, X. Xiao, Z. Lin, H. Chen, Z. Niu,  An extensive benchmark study on biomedical text generation and mining with ChatGPT,  Bioinformatics 39 (2023) btad557. URL: https://doi.org/10.1093/bioinformatics/btad557. doi:10.1093/bioinformatics/btad557.
  * Tian et al. [2024] S. Tian, Q. Jin, L. Yeganova, P.-T. Lai, Q. Zhu, X. Chen, Y. Yang, Q. Chen, W. Kim, D. C. Comeau, R. Islamaj, A. Kapoor, X. Gao, Z. Lu,  Opportunities and challenges for ChatGPT and large language models in biomedicine and health,  Briefings in Bioinformatics 25 (2024) bbad493. URL: https://doi.org/10.1093/bib/bbad493. doi:10.1093/bib/bbad493.
  * OpenAI [2023] OpenAI,  Gpt-3.5 turbo,  OpenAI (2023). URL: https://platform.openai.com/docs/models/gpt-3-5-turbo.
  * Touvron et al. [2023] H. Touvron, T. Lavril, G. Izacard, X. Martinet, M.-A. Lachaux, T. Lacroix, B. Rozière, N. Goyal,  Llama open and efficient foundation language models,  arXiv preprint arXiv:2302.13971 (2023). URL: https://arxiv.org/abs/2302.13971.
  * Neumann et al. [2019] M. Neumann, D. King, I. Beltagy, W. Ammar,  ScispaCy: Fast and Robust Models for Biomedical Natural Language Processing (2019) 319–327. URL: https://www.aclweb.org/anthology/W19-5034. doi:10.18653/v1/W19-5034. arXiv:arXiv:1902.07669.
  * Bodenreider [2004] O. Bodenreider,  The unified medical language system (umls): integrating biomedical terminology,  Nucleic acids research 32 (2004) D267–D270. URL: https://doi.org/10.1093/nar/gkh061. doi:10.1093/nar/gkh061.
  * uml [2023] Umls terminology services (2023). URL: https://uts.nlm.nih.gov/uts/umls/home.
  * Fenogenova et al. [2024] A. Fenogenova, A. Chervyakov, N. Martynov, A. Kozlova, M. Tikhonova, A. Akhmetgareeva, A. Emelyanov, D. Shevelev, P. Lebedev, L. Sinev, U. Isaeva, K. Kolomeytseva, D. Moskovskiy, E. Goncharova, N. Savushkin, P. Mikhailova, D. Dimitrov, A. Panchenko, S. Markov, Mera: A comprehensive llm evaluation in russian, 2024. URL: https://arxiv.org/abs/2401.04531. arXiv:2401.04531.
  * Grammarly [2024] Grammarly,  Grammarly handbook (2024). URL: https://www.grammarly.com/handbook/.


# On the quantum origin of potentials

Saurya Das<EMAIL_ADDRESS>Theoretical Physics Group, Department of
Physics and Astronomy, University of Lethbridge, 4401 University Drive,
Lethbridge, Alberta T1K 3M4, Canada  Sourav Sur<EMAIL_ADDRESS>Department of Physics and Astrophysics, University of Delhi, New Delhi 110007,
India

###### Abstract

The dynamics of a quantum particle is governed by its wavefunction, which in
turn is determined by the classical potential to which it is subjected.
However the wavefunction itself induces a quantum potential, the particle
‘sees’ the sum of the classical and quantum potentials, and there is no way to
separate the two. Therefore in principle, part or whole of an observed
potential may be attributable to a quantum potential. We examine this
possibility and discuss implications.

The standard paradigm of quantum mechanics, non-relativistic or relativistic,
consists of solving a suitable wave equation (such as Schrödinger, Klein-
Gordon or Dirac) with an appropriate potential, for normalizable
wavefunctions. However, there is a lesser known result that the wavefunction
of a quantum particle itself generates a ‘quantum potential’ and the dynamics
of the particle is governed by the sum of the classical and quantum potentials
and not by them individually. Therefore, part or all of one can be replaced by
the other, and in principle it is possible that particles are intrinsically in
a flat spacetime without any potentials and what is normally perceived as a
classical potential, for e.g. a harmonic oscillator, Coulomb or the
gravitational potential, is in reality, a quantum potential generated by the
wavefunction of the particle.

To explore this possibility, for simplicity, let us start with the Schrödinger
equation (generalization to relativistic wave equations should be
straightforward)

$-\,\frac{\hbar^{2}}{2m}\,\nabla^{2}\Psi+\,V\Psi=\,i\,\frac{\partial\Psi}{\partial
t}~{},$ (1)

with the wavefunction written as

$\Psi(\vec{\boldsymbol{x}},t)=\,\mathcal{R}(\vec{\boldsymbol{x}},t)\,e^{i\,\mathcal{S}(\vec{\boldsymbol{x}},t)}\,\quad\left[\mathcal{R},\mathcal{S}\in\mathbb{R}\right]\,.$
(2)

Substituting this back in Eq. (1), and extracting the real and imaginary
parts, yields two equations bohm

$\displaystyle\vec{\nabla}\cdot\vec{\boldsymbol{J}}+\,\frac{\partial\rho}{\partial
t}=\,0\,,$ (3) $\displaystyle
m\,\frac{d\vec{v}}{dt}=\,-\,\vec{\nabla}\left(V+\,V_{Q}\right)=-\vec{\nabla}V_{tot}\,,$
(4)

where
$\rho=|\Psi|^{2}\,,~{}\vec{\boldsymbol{J}}=\frac{\hbar}{2mi}\left[\Psi^{\star}\vec{\nabla}\Psi-\Psi\vec{\nabla}\Psi^{\star}\right]$
are respectively the probability density and the probability current density,
$\vec{v}:=(\hbar/m)\,\vec{\nabla}\mathcal{S}\,$ is the ‘velocity field’,

$V_{Q}=\,-\,\frac{\hbar^{2}}{2m}\frac{\nabla^{2}\mathcal{R}}{\mathcal{R}}\,,$
(5)

is the ‘quantum potential’, and

$V_{tot}\equiv V+V_{Q}~{}.$ (6)

Eq. (3) is simply the standard conservation equation for $\rho$ and
$\vec{\boldsymbol{J}}$. Eq. (4), on the other hand, shows that the quantum
dynamics of a particle of mass $m$ is still governed by the classical Newton’s
equation, albeit with the classical potential $V$ augmented by the
wavefunction dependent ‘quantum potential’. For instance, while a stream of
free electrons, governed just by a vanishing classical potential ($V=0$),
follows straight line trajectories, that governed by $V_{tot}=V+V_{Q}$, for an
appropriate choice of $\Psi(\vec{\boldsymbol{x}},t)$ reproduces the familiar
interference patterns fringes . Thus, we see that a quantum particle subjected
to $V_{tot}$ has no way of figuring out what part of it is ‘classical’ and
what part ‘quantum’, and a part (or all) of the classical potential may in
fact be the quantum potential generated by a suitable wavefunction and vice-
versa. Going one step further, one can even say that observed potentials and
wavefunctions are interchangeable and it may be a matter of choice as to which
one is considered more fundamental.

Before proceeding further, we present an even lesser known result which
follows from Eqs. (1) and (5), namely, that for a stationary state, of energy
$E$ and described by the wavefunction
$\,\Psi(\vec{\boldsymbol{x}},t)=\psi(\vec{\boldsymbol{x}})\,e^{-iEt/\hbar}\,$,
one finds111 The easiest way is to see this is to recognize that
$\nabla^{2}{\cal R}/{\cal R}=\nabla^{2}{\psi}/{\psi}$ for stationary states,
and substituting for $\nabla^{2}\psi$ using the time-independent Schrödinger
equation $-\,\frac{\hbar^{2}}{2m}\,\nabla^{2}\psi+\,V\psi=\,E\psi\,$ in
Eq.(5).

$V_{Q}=\,-\,V+\,E\,.$ (7)

The result is remarkable for two reasons. First, it shows that the $\hbar$’s
cancel out from Eq. (5) and the induced quantum potential is simply equal and
opposite to the starting classical potential $V$ (up to an unimportant
constant). Accordingly, the quantum force is also equal and opposite to the
starting classical force. Second, since as mentioned above, $V$ and $V_{Q}$
are interchangeable, it follows that if we had started with $V_{Q}$ instead of
$V$ in the Schrödinger equation, it would have induced an equal and opposite
$V$ via quantum mechanics! In other words, what is to be treated as classical
and what as quantum may just be a matter of choice, and the presence of any
one would cause the other to emerge naturally, and this is true regardless of
the exact form of the starting potential. For example, if one starts with an
attractive gravitational potential, one would end up with a repulsive quantum
potential. Alternatively, the observed attractive gravitational potential is
equally likely to be, in reality, the quantum potential induced by a repulsive
classical potential sd ; db1 ; db2 ; perelman ; dsemergent !

Extending the idea further, we now ask a natural question: what if a particle
experiences a potential, but the starting classical potential is zero, i.e.
the particle is in free space, albeit with a wavefunction whose amplitude
satisfies Eq. (5)? Evidently, the following form of Eq. (5)

${\nabla^{2}\mathcal{R}}+\frac{2m\,V_{Q}}{\hbar^{2}}\,{\cal R}=0~{}$ (8)

makes it clear that the particle experiences a potential $V_{Q}$. However,
there is no way to tell whether this is a background potential, or that
resulting from a specific form of the real part of the wave-amplitude (to be
considered as time-independent and denoted by
$\mathcal{R}(\vec{\boldsymbol{x}})$). Note that this does not completely
specify the full wavefunction $\Psi(\vec{\boldsymbol{x}},t)$, as its phase
$\mathcal{S}(\vec{\boldsymbol{x}},t)$ remains undetermined. Nevertheless, the
validity of the standard probability interpretation of quantum mechanics
requires $\Psi(\vec{\boldsymbol{x}},t)$, and hence
$\mathcal{R}(\vec{\boldsymbol{x}})$, to be normalizable in the first place.
Therefore, one can take the route of finding normalizable solutions of Eq. (8)
for a host of potentials. We however adopt a simpler approach in which $V_{Q}$
in Eq. (8) is replaced by $V_{Q}+E_{0}$, where for simplicity, we take $E_{0}$
as the ground state energy of the system under consideration. So, the Eq. (8)
becomes

${\nabla^{2}\mathcal{R}}+\frac{2m\,\left[E_{0}-(-V_{Q})\right]}{\hbar^{2}}\,{\cal
R}=0\,.$ (9)

This is nothing but the time-independent Schrödinger equation corresponding to
the ground state, but for a potential $-V_{Q}$. However, the interpretation is
quite different here. Let us assume that there exist normalizable solutions of
this equation, for some reasonable potentials. Now, the requirement of
$\mathcal{R}$ to be real is guaranteed at least for the ground state realpsi .
We can therefore stipulate the potential $-V_{Q}-E_{0}$ (or approximately,
$-V_{Q}$, for $|V_{Q}|\gg|E_{0}|$) to be that felt effectively by a particle,
provided the amplitude of the particle’s wavefunction, naturally arising or
otherwise, is a solution of Eq. (9).

As an example, consider an inverted linear harmonic oscillator and postulate
that the ‘observed’ potential $V=-m\,\omega^{2}x^{2}/2\,$ is in reality the
quantum potential experienced by a free particle, described by an wave-
function whose amplitude satisfies Eq. (9) with $V_{Q}=V$, that now reads

${\nabla^{2}\mathcal{R}}+\,\frac{2m}{\hbar^{2}}\left(E_{0}-\frac{m\omega^{2}\,x^{2}}{2}\right){\cal
R}=\,0\,.$ (10)

This is just the stationary state Schrödinger equation for a linear harmonic
oscillator potential. The solution is of course the unique, normalizable
ground state wavefunction given by

$\displaystyle{\cal
R}=\left(\frac{m\omega}{\pi\,\hbar}\right)^{1/4}\,e^{-m\omega
x^{2}/2\,\hbar}\,.$ (11)

Similarly, substituting for a repulsive Coulomb potential $V_{Q}=e^{2}/r$, and
replacing it with $(V_{Q}+E_{0})$ in Eq. (9), where
$E_{0}=-me^{4}/(2\,\hbar^{2})$ is the ground state energy of the Hydrogen
atom, one finds the unique normalizable solution

$\displaystyle\mathcal{R}=\,\frac{a_{0}^{-3/2}}{\sqrt{\pi}}\,e^{-r/a_{0}}~{},$
(12)

with $a_{0}=\hbar^{2}/(me^{2})$. One may carry on this way by considering any
other potential, and deriving the corresponding $\mathcal{R}$ via Eq. (9).

Figure 1: Quantum potential $V_{Q}=-m\omega^{2}x^{2}/2$ and its source
wavefunction $\mathcal{R}$, for the fiducial settings $m=1,\omega=1/2$.
Figure 2: Quantum potential $V_{Q}=e^{2}/r$ and its source wavefunction
$\mathcal{R}$, for the fiducial settings $e=1,m=0.511$.

To test the robustness of the method, we consider a couple of more examples
and obtain the wavefunction required to produce a given quantum potential,
using Eq.(8) directly. First, the ‘step potential’ (in one-dimension), defined
by $V_{Q}=0$ for $x<0$, and $V_{Q}=V_{0}$ for $x\geq 0$, where $V_{0}$ is a
constant. The corresponding wavefunction is given by $\mathcal{R}=1$ for
$x<0$, and $\mathcal{R}=\cos(kx)$ for $x\geq 0$, where
$k=\sqrt{2mV_{0}/\hbar^{2}}$. It can be easily verified that Eq. (5) gives the
required step potential.

The last example we consider is a linear potential, also in one dimensions,
given by $V_{Q}=\kappa x$, where $\kappa$ is a constant. In this case,
substituting in Eq. (5) and solving for $\mathcal{R}$, we get
$\mathcal{R}=Ai(-k_{1}^{1/3}\,x)$ or $\mathcal{R}=Bi(-k_{1}^{1/3}\,x)$, where
$Ai$ and $Bi$ are the Airy functions and $k_{1}=2m\kappa/\hbar^{2}$.

The above examples emphasize the suitability of the method, although we note
that the number of exactly solvable examples from Eq. (5) is limited, and one
would have to resort to numerical techniques for more complicated potentials.
The plots of the various quantum potentials and the corresponding
wavefunctions that give rise to them are shown in the Figures 1-4.

Figure 3: Quantum potential $V_{Q}=0~{}(x<0)$ and $V_{Q}=V_{0}~{}(x\geq 0)$
and its source wavefunction $\mathcal{R}$, for the fiducial settings
$m=1,V_{0}=1.5$. Figure 4: Quantum potential $V_{Q}=\kappa\,x$ and its source
wavefunction $\mathcal{R}$, for the fiducial settings $m=1,\kappa=0.1$.

To summarize, we have shown that practically any presumptive classical
potential can in fact be derived as the quantum potential of a suitable
wavefunction. These could be familiar potentials such as the Coulomb, inverted
harmonic oscillator or the gravitational potential, or exotic ones such as
those which can effectively explain the flat galaxy rotation curves. Therefore
it begs the questions as to whether the ones we observe as potentials and
interactions are mere manifestations of appropriate wavefunctions. The
question is particularly important in the context of the gravitational field,
since the corresponding field theory is known to be non-renormalizable.
Therefore, a more fundamental picture, with the observed gravitational fields
as quantum potentials of particle wavefunctions would offer a fresh insight
into the problem. It should even predict corrections to the $1/r$ potential,
at very small or large length scales, beyond the regimes in which the
Newtonian gravity has been verified experimentally. Of course, there are many
questions to be answered. Some of these should be straightforward, for
example, the relativistic generalization of our work starting from a
relativistic wave equation (part of it has been done in ref. sd ). The others,
for instance, the determination of the phase $\mathcal{S}$ and how the
appropriate wavefunctions can be specified, require further studies.

To conclude, we list a few important observations and possible directions for
future works.

1\. Excited states: we remind that Eq. (9) is an approximation of Eq. (8),
where the ground state energy $E_{0}$ was introduced in an attempt to
guarantee a normalizable $\mathcal{R}$, albeit at the cost of ‘raising’ the
quantum potential by a small amount. Similarly, one may replace
$E_{0}\rightarrow E_{n}$, an excited state energy, and
$\mathcal{R}\rightarrow\mathcal{R}_{n}$, the corresponding excited state. This
demonstrates the non-uniqueness of the wavefunction in general for a given
quantum potential.

2\. Non-stationary states: as Eq. (8) suggests, a time-dependent quantum
potential $V_{Q}$ would require a time-dependent $\mathcal{R}$ as its source.
Furthermore, one sees from Eq. (3) that such a $\mathcal{R}$ would entail a
non-trivial phase $S$.

3\. Other formulations: It may be noted that a ‘modified potential’, distinct
from, but related to the quantum potential, was defined and its implications
explored in detail in floyd . A re-examination of our results in light of the
aforementioned potential may shed further light on emergent potentials.
Similarly, the possible origin of the Schrödinger equation purely from
symmetry considerations was examined in faraggi1 ; faraggi2 . That study may
also complement the current work, as it provides a justification for starting
with the Schrödinger equation in the first place. It may also be useful in
extending our results to other quantum wave equations.

4\. Future directions: in addition to studying the above issues, namely
excited states, non-stationary states and other formalisms, we will carefully
investigate a few other things. This includes, the (special) relativistic
generalization of our results, computing corrections to the gravitational
potential at various length scales, studying the stability of the
wavefunction, generalization to higher dimensions and last but not the least,
examine whether the idea can be generalized to Yang-Mills potential. We hope
to report on these elsewhere.

Acknowledgment

This work was supported by the Natural Sciences and Engineering Research
Council of Canada. We thank the anonymous referee for useful comments which
have helped us to improve the manuscript.

## References

  * (1) D. Bohm, Phys. Rev. 85 (1952) 166; D. Bohm, B. J. Hiley, P. N. Kaloyerou, Phys. Rep. 144, no.6 (1987) 321.
  * (2) C. Phillipides, C. Dewdney, B. J. Hiley, il nuovo cim. B 52, no.1 (1979) 15.
  * (3) S. Das, Phys. Rev. D 89 (2014) 084068 [arXiv:1311.6539].
  * (4) S. Das, R. K. Bhaduri, Class. Quant. Grav. 32 (2015) 105003 [arXiv1411.0753].
  * (5) S. Das, R. K. Bhaduri, Phys. News 40 (2019) 81 [arXiv:1808.10505].
  * (6) C. C. Perelman, Phys. Lett. B 788 (2019) 546.
  * (7) S. Das, S. Sur, arXiv:2105.07150.
  * (8) J. Mur-Petit, A. Polls, F. Mazzanti, Am. J. Phys. 70(8) (2002) 808.
  * (9) E. R. Floyd, Phys. Rev. D 26 (1982) 1339.
  * (10) A. E. Faraggi and M. Matone, Phys. Lett. B 450 (1999) 34.
  * (11) A. E. Faraggi and M. Matone, Int. J. Mod. Phys. A 15 (2000) 1869.


stochastic dynamics at any order, with an explicit expression for the modified
vector field.

There are a handful of challenging open questions that follow from the present
work. The deeper understanding of the integration by parts operation given in
Proposition 3.4 in the Euclidean setting is key in extending Theorems 3.10 and
3.14 to the manifold case for the creation of modified equations and the
backward error analysis of projection methods. An explicit description of the
combination of forests that vanish by integration by parts (see Remark 3.5) is
necessary for the derivation of order conditions for the invariant measure at
any order. Following Remark 3.12, an alternate approach could be to
characterise the modified vector field of methods that preserve the invariant
measure exactly in the spirit of [55, 54] in the context of volume-preserving
methods. The exotic aromatic formalism is the natural tool in this context as
the operations $\operatorname{div}$ and $\langle f,{-}\rangle$ of equation
(10) precisely generate the aromas. This calls for future works in the spirit
of [55, 54].

The Butcher forests and their extensions have been successfully applied to
rough paths with the construction of branched rough paths [40] (see also [43,
38]), planarly branched rough paths [78, 29, 34], and aromatic rough paths
[61]. In this context, the substitution law corresponds to the translation of
rough paths, and the notion of composition of exotic aromatic trees [76, 56]
is closely linked to the Hopf algebra of multi-indices [65, 48]. The creation
and study of rough paths structures arising from exotic aromatic or exotic
clumped forests is natural and is matter for later work. Moreover, we use the
clumping idea to explain the algebraic behaviour brought by the divergence and
the scalar product operators. One could consider general multilinear maps that
transform trees into rootless graphs. The questions of the freeness, the
general algebraic structure, and the potential applications are all
interesting matters for follow ups.

Following the recent work [10], the approach with projection methods for the
sampling of ergodic dynamics on manifolds could be replaced with an intrinsic
approach with Lie-group methods. This new approach could yield simple and
efficient intrinsic discretisations of high-order for sampling the invariant
measure, which could be combined with other popular techniques such as
postprocessing, Metropolisation,…It could also greatly simplify the tedious
algebraic structure associated to projection methods and yield a structure
similar to the ones appearing in the study of Lie group methods [46]. This
will be studied in upcoming works.

Acknowledgements. The authors would like to thank Dominique Manchon, Hans
Munthe-Kaas, and Gilles Vilmart for helpful discussions. The authors
acknowledge the support of the Swiss National Science Foundation, projects No
200020_214819, and No. 200020_192129, of the Research Council of Norway
through project 302831 “Computational Dynamics and Stochastics on Manifolds”
(CODYSMA), and of the French program ANR-11-LABX-0020-0 (Labex Lebesgue).

## References

  * [1] A. Abdulle, I. Almuslimani, and G. Vilmart.  Optimal explicit stabilized integrator of weak order 1 for stiff and ergodic stochastic differential equations.  SIAM/ASA J. Uncertain. Quantif., 6(2):937–964, 2018.
  * [2] A. Abdulle, D. Cohen, G. Vilmart, and K. C. Zygalakis.  High weak order methods for stochastic differential equations based on modified equations.  SIAM J. Sci. Comput., 34(3):A1800–A1823, 2012.
  * [3] A. Abdulle, G. Vilmart, and K. C. Zygalakis.  High order numerical approximation of the invariant measure of ergodic SDEs.  SIAM J. Numer. Anal., 52(4):1600–1622, 2014.
  * [4] A. Abdulle, G. Vilmart, and K. C. Zygalakis.  Long time accuracy of Lie-Trotter splitting methods for Langevin dynamics.  SIAM J. Numer. Anal., 53(1):1–16, 2015.
  * [5] I. M. Anderson.  The variational bicomplex.  Unpublished, 1989.
  * [6] I. M. Anderson.  Introduction to the variational bicomplex.  In Mathematical aspects of classical field theory (Seattle, WA, 1991), volume 132 of Contemp. Math., pages 51–73. Amer. Math. Soc., Providence, RI, 1992.
  * [7] I. Angiono, C. Galindo, and L. Vendramin.  Hopf braces and Yang-Baxter operators.  Proc. Amer. Math. Soc., 145(5):1981–1995, 2017.
  * [8] S. Anmarkrud and A. Kværnø.  Order conditions for stochastic Runge-Kutta methods preserving quadratic invariants of Stratonovich SDEs.  J. Comput. Appl. Math., 316:40–46, 2017.
  * [9] G. Benettin and A. Giorgilli.  On the Hamiltonian interpolation of near-to-the-identity symplectic mappings with application to symplectic integration algorithms.  J. Statist. Phys., 74(5-6):1117–1143, 1994.
  * [10] K. Bharath, A. Lewis, A. Sharma, and M. V. Tretyakov.  Sampling and estimation on manifolds using the Langevin diffusion.  arXiv preprint arXiv:2312.14882, 2023.
  * [11] G. Bogfjellmo.  Algebraic structure of aromatic B-series.  J. Comput. Dyn., 6(2):199–222, 2019.
  * [12] G. Bogfjellmo, E. Celledoni, R. I. McLachlan, B. Owren, and G. R. W. Quispel.  Using aromas to search for preserved measures and integrals in Kahan’s method.  Math. Comp., 93(348):1633–1653, 2024.
  * [13] N. Bou-Rabee and H. Owhadi.  Long-run accuracy of variational integrators in the stochastic context.  SIAM J. Numer. Anal., 48(1):278–297, 2010.
  * [14] E. Bronasco.  Exotic B-series and S-series: algebraic structures and order conditions for invariant measure sampling.  Found. Comput. Math., pages 1–31, 2024.
  * [15] Y. Bruned and D. Manchon.  Algebraic deformation for (S)PDEs.  J. Math. Soc. Japan, 75(2):485–526, 2023.
  * [16] K. Burrage and P. M. Burrage.  High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations.  Appl. Numer. Math., 22(1-3):81–101, 1996.  Special issue celebrating the centenary of Runge-Kutta methods.
  * [17] K. Burrage and P. M. Burrage.  Order conditions of stochastic Runge-Kutta methods by B-series.  SIAM J. Numer. Anal., 38(5):1626–1646, 2000.
  * [18] J. C. Butcher.  The effective order of Runge-Kutta methods.  In Conf. on Numerical Solution of Differential Equations (Dundee, 1969), pages 133–139. Springer, Berlin, 1969.
  * [19] J. C. Butcher.  An algebraic theory of integration methods.  Math. Comp., 26:79–106, 1972.
  * [20] J. C. Butcher.  Numerical methods for ordinary differential equations.  John Wiley & Sons, Ltd., Chichester, third edition, 2016.
  * [21] J. C. Butcher.  B-series: algebraic analysis of numerical methods.  Springer, 2021.
  * [22] D. Calaque, K. Ebrahimi-Fard, and D. Manchon.  Two interacting Hopf algebras of trees: a Hopf-algebraic approach to composition and substitution of B-series.  Adv. in Appl. Math., 47(2):282–308, 2011.
  * [23] M. P. Calvo, A. Murua, and J. M. Sanz-Serna.  Modified equations for ODEs.  In Chaotic numerics (Geelong, 1993), volume 172 of Contemp. Math., pages 63–74. Amer. Math. Soc., Providence, RI, 1994.
  * [24] P. Chartier, E. Hairer, and G. Vilmart.  A substitution law for B-series vector fields.  Research Report RR-5498, INRIA, 2005.
  * [25] P. Chartier, E. Hairer, and G. Vilmart.  Numerical integrators based on modified differential equations.  Math. Comp., 76(260):1941–1953, 2007.
  * [26] P. Chartier, E. Hairer, and G. Vilmart.  Algebraic structures of B-series.  Found. Comput. Math., 10(4):407–427, 2010.
  * [27] P. Chartier and A. Murua.  Preserving first integrals and volume forms of additively split systems.  IMA J. Numer. Anal., 27(2):381–405, 2007.
  * [28] A. Connes and D. Kreimer.  Hopf algebras, renormalization and noncommutative geometry.  Comm. Math. Phys., 199(1):203–242, 1998.
  * [29] C. Curry, K. Ebrahimi-Fard, D. Manchon, and H. Z. Munthe-Kaas.  Planarly branched rough paths and rough differential equations on homogeneous spaces.  J. Differential Equations, 269(11):9740–9782, 2020.
  * [30] K. Debrabant.  Runge-Kutta methods for third order weak approximation of SDEs with multidimensional additive noise.  BIT Numer. Math., 50(3):541–558, 2010.
  * [31] K. Debrabant and A. Kværnø.  B-series analysis of stochastic Runge-Kutta methods that use an iterative scheme to compute their internal stage values.  SIAM J. Numer. Anal., 47(1):181–203, 2008/09.
  * [32] K. Debrabant and A. Kværnø.  Composition of stochastic B-series with applications to implicit Taylor methods.  Appl. Numer. Math., 61(4):501–511, 2011.
  * [33] A. Debussche and E. Faou.  Weak backward error analysis for SDEs.  SIAM J. Numer. Anal., 50(3):1735–1752, 2012.
  * [34] K. Ebrahimi-Fard and L. Rahm.  Primitive elements in the Munthe-Kaas-Wright Hopf algebra.  Submitted, 2023.
  * [35] E. Faou and T. Lelièvre.  Conservative stochastic differential equations: mathematical and numerical analysis.  Math. Comp., 78(268):2047–2074, 2009.
  * [36] G. Fløystad, D. Manchon, and H. Z. Munthe-Kaas.  The universal pre-Lie-Rinehart algebras of aromatic trees.  In Geometric and harmonic analysis on homogeneous spaces and applications, volume 366 of Springer Proc. Math. Stat., pages 137–159. Springer, Cham, [2021] ©2021.
  * [37] L. Foissy.  Algebraic structures on typed decorated rooted trees.  SIGMA Symmetry Integrability Geom. Methods Appl., 17:Paper No. 086, 28, 2021.
  * [38] P. K. Friz and M. Hairer.  A course on rough paths.  Universitext. Springer, Cham, [2020] ©2020.  With an introduction to regularity structures, Second edition of [3289027].
  * [39] S. D. Giovacchino, D. J. Higham, and K. Zygalakis.  Backward error analysis and the qualitative behaviour of stochastic optimization algorithms: Application to stochastic coordinate descent.  Submitted, 2023.
  * [40] M. Gubinelli.  Ramification of rough paths.  J. Differential Equations, 248(4):693–721, 2010.
  * [41] E. Hairer, C. Lubich, and G. Wanner.  Geometric numerical integration, volume 31 of Springer Series in Computational Mathematics.  Springer-Verlag, Berlin, second edition, 2006.  Structure-preserving algorithms for ordinary differential equations.
  * [42] E. Hairer and G. Wanner.  On the Butcher group and general multi-value methods.  Computing (Arch. Elektron. Rechnen), 13(1):1–15, 1974.
  * [43] M. Hairer and D. Kelly.  Geometric versus non-geometric rough paths.  Ann. Inst. Henri Poincaré Probab. Stat., 51(1):207–251, 2015\.
  * [44] R. Z. Hasminskii.  Stochastic stability of differential equations, volume 7 of Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis.  Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980.  Translated from the Russian by D. Louvish.
  * [45] M. E. Hoffman.  Combinatorics of rooted trees and Hopf algebras.  Trans. Amer. Math. Soc., 355(9):3795–3811, 2003.
  * [46] A. Iserles, H. Z. Munthe-Kaas, S. P. Nørsett, and A. Zanna.  Lie-group methods.  In Acta numerica, 2000, volume 9 of Acta Numer., pages 215–365. Cambridge Univ. Press, Cambridge, 2000.
  * [47] A. Iserles, G. R. W. Quispel, and P. S. P. Tse.  B-series methods cannot be volume-preserving.  BIT Numer. Math., 47(2):351–378, 2007.
  * [48] J.-D. Jacques and L. Zambotti.  Post-Lie algebras of derivations and regularity structures.  Submitted, arXiv:2306.02484, 2023.
  * [49] Y. Komori, T. Mitsui, and H. Sugiura.  Rooted tree analysis of the order conditions of ROW-type scheme for stochastic differential equations.  BIT Numer. Math., 37(1):43–66, 1997.
  * [50] M. Kopec.  Weak backward error analysis for Langevin process.  BIT Numer. Math., 55(4):1057–1103, 2015.
  * [51] M. Kopec.  Weak backward error analysis for overdamped Langevin processes.  IMA J. Numer. Anal., 35(2):583–614, 2015.
  * [52] D. Küpper, A. Kværnø, and A. Rößler.  A Runge-Kutta method for index 1 stochastic differential-algebraic equations with scalar noise.  BIT Numer. Math., 52(2):437–455, 2012.
  * [53] A. Laurent.  Algebraic Tools and Multiscale Methods for the Numerical Integration of Stochastic Evolutionary Problems.  PhD thesis, University of Geneva, 2021.
  * [54] A. Laurent.  The Lie derivative and Noether’s theorem on the aromatic bicomplex for the study of volume-preserving numerical integrators.  J. Comput. Dyn., 11(1):10–22, 2024.
  * [55] A. Laurent, R. I. McLachlan, H. Z. Munthe-Kaas, and O. Verdier.  The aromatic bicomplex for the description of divergence-free aromatic forms and volume-preserving integrators.  Forum Math. Sigma, 11:Paper No. e69, 2023.
  * [56] A. Laurent and H. Z. Munthe-Kaas.  The universal equivariance properties of exotic aromatic B-series.  Submitted, arXiv:2305.10993, 2023.
  * [57] A. Laurent and G. Vilmart.  Exotic aromatic B-series for the study of long time integrators for a class of ergodic SDEs.  Math. Comp., 89(321):169–202, 2020.
  * [58] A. Laurent and G. Vilmart.  Order conditions for sampling the invariant measure of ergodic stochastic differential equations on manifolds.  Found. Comput. Math., 22(3):649–695, 2022.
  * [59] B. Leimkuhler and C. Matthews.  Rational construction of stochastic numerical methods for molecular sampling.  Appl. Math. Res. Express. AMRX, 2013(1):34–56, 2013.
  * [60] B. Leimkuhler, C. Matthews, and G. Stoltz.  The computation of averages from equilibrium and nonequilibrium Langevin molecular dynamics.  IMA J. Numer. Anal., 36(1):13–79, 2016.
  * [61] A. Lejay.  Constructing general rough differential equations through flow approximations.  Electron. J. Probab., 27:Paper No. 7, 24, 2022.
  * [62] T. Lelièvre, M. Rousset, and G. Stoltz.  Free energy computations.  Imperial College Press, London, 2010.  A mathematical perspective.
  * [63] Q. Li, C. Tai, and W. E.  Stochastic modified equations and dynamics of stochastic gradient algorithms I: mathematical foundations.  J. Mach. Learn. Res., 20:Paper No. 40, 47, 2019.
  * [64] Y. Li, Y. Sheng, and R. Tang.  Post-Hopf algebras, relative Rota-Baxter operators and solutions of the Yang-Baxter equation.  Journal of Noncommutative Geometry, 2023.
  * [65] P. Linares, F. Otto, and M. Tempelmayr.  The structure group for quasi-linear equations via universal enveloping algebras.  Comm. Amer. Math. Soc., 3:1–64, 2023.
  * [66] J.-L. Loday and M. Ronco.  Combinatorial Hopf algebras.  In Quanta of maths, volume 11 of Clay Math. Proc., pages 347–383. Amer. Math. Soc., Providence, RI, 2010.
  * [67] J.-H. Lu.  Hopf algebroids and quantum groupoids.  Internat. J. Math., 7(1):47–70, 1996.
  * [68] A. Lundervold and H. Munthe-Kaas.  Hopf algebras of formal diffeomorphisms and numerical integration on manifolds.  In Combinatorics and physics, volume 539 of Contemp. Math., pages 295–324. Amer. Math. Soc., Providence, RI, 2011.
  * [69] A. Lundervold and H. Munthe-Kaas.  Backward error analysis and the substitution law for Lie group integrators.  Found. Comput. Math., 13(2):161–186, 2013.
  * [70] A. Lundervold and H. Z. Munthe-Kaas.  On algebraic structures of numerical integration on vector spaces and manifolds.  In Faà di Bruno Hopf algebras, Dyson-Schwinger equations, and Lie-Butcher series, volume 21 of IRMA Lect. Math. Theor. Phys., pages 219–263. Eur. Math. Soc., Zürich, 2015.
  * [71] D. Manchon and A. Saïdi.  Lois pré-Lie en interaction.  Comm. Algebra, 39(10):3662–3680, 2011.
  * [72] R. I. McLachlan, K. Modin, H. Munthe-Kaas, and O. Verdier.  B-series methods are exactly the affine equivariant methods.  Numer. Math., 133(3):599–622, 2016.
  * [73] R. I. McLachlan, K. Modin, H. Munthe-Kaas, and O. Verdier.  Butcher series: a story of rooted trees and numerical methods for evolution equations.  Asia Pac. Math. Newsl., 7(1):1–11, 2017.
  * [74] G. N. Milstein and M. V. Tretyakov.  Stochastic numerics for mathematical physics.  Scientific Computation. Springer-Verlag, Berlin, 2004.
  * [75] I. Moerdijk and J. Mrčun.  On the universal enveloping algebra of a Lie algebroid.  Proc. Amer. Math. Soc., 138(9):3135–3145, 2010.
  * [76] H. Munthe-Kaas and O. Verdier.  Aromatic Butcher series.  Found. Comput. Math., 16(1):183–215, 2016.
  * [77] H. Z. Munthe-Kaas and W. M. Wright.  On the Hopf algebraic structure of Lie group integrators.  Found. Comput. Math., 8(2):227–257, 2008.
  * [78] H. Z. Munthe-Kaas and W. M. Wright.  On the Hopf algebraic structure of Lie group integrators.  Found. Comput. Math., 8(2):227–257, 2008.
  * [79] J.-M. Oudom and D. Guin.  On the Lie enveloping algebra of a pre-Lie algebra.  J. K-Theory, 2(1):147–167, 2008.
  * [80] L. Rahm.  An operadic approach to substitution in Lie-Butcher series.  Forum Math. Sigma, 10:Paper No. e20, 29, 2022.
  * [81] A. Rößler.  Stochastic Taylor expansions for the expectation of functionals of diffusion processes.  Stochastic Anal. Appl., 22(6):1553–1576, 2004.
  * [82] A. Rößler.  Rooted tree analysis for order conditions of stochastic Runge-Kutta methods for the weak approximation of stochastic differential equations.  Stoch. Anal. Appl., 24(1):97–134, 2006.
  * [83] A. Rößler.  Runge-Kutta methods for Itô stochastic differential equations with scalar noise.  BIT Numer. Math., 46(1):97–110, 2006.
  * [84] A. Rößler.  Stochastic Taylor expansions for functionals of diffusion processes.  Stoch. Anal. Appl., 28(3):415–429, 2010.
  * [85] A. Saïdi.  On a pre-Lie algebra defined by insertion of rooted trees.  Lett. Math. Phys., 92(2):181–196, 2010.
  * [86] A. Saïdi.  Algèbres de Hopf d’arbres et structures pré-Lie.  PhD thesis, Université Blaise Pascal-Clermont-Ferrand II, 2011.
  * [87] J. M. Sanz-Serna.  Markov chain Monte Carlo and numerical differential equations.  In Current challenges in stability issues for numerical differential equations, volume 2082 of Lecture Notes in Math., pages 39–88. Springer, Cham, 2014.
  * [88] T. Shardlow.  Modified equations for stochastic differential equations.  BIT Numer. Math., 46(1):111–125, 2006.
  * [89] D. Talay and L. Tubaro.  Expansion of the global error for numerical schemes solving stochastic differential equations.  Stochastic Anal. Appl., 8(4):483–509 (1991), 1990.
  * [90] G. Vilmart.  Postprocessed integrators for the high order integration of ergodic SDEs.  SIAM J. Sci. Comput., 37(1):A201–A220, 2015.
  * [91] K. C. Zygalakis.  On the existence and the applications of modified equations for stochastic differential equations.  SIAM J. Sci. Comput., 33(1):102–130, 2011.

##  Appendix A Examples for forests of small order

We present in Tables LABEL:appendix:BCK_coproduct and
LABEL:appendix:CEM_coaction the Butcher-Connes-Kreimer coproduct and the
substitution law of all primitive exotic aromatic forests
$\operatorname{Prim}(\mathcal{EAF},\Delta_{\mathcal{EA}})$ of order up to
three. To compute the Butcher-Connes-Kreimer coproduct efficiently, we use
Remark 4.7 which implies the property

$\Delta_{BCK}(\pi_{1}\cdot\pi_{2})=\Delta_{BCK}(\pi_{1})\hat{\cdot}\Delta_{BCK}(\pi_{2}),$

where $(x\otimes y)\hat{\cdot}(u\cdot v)=(x\cdot u)\otimes(y\cdot v)$ and
$\pi_{1},\pi_{2}\in\mathcal{AF}_{D}$. Therefore, the values of the Butcher-
Connes-Kreimer coproduct on primitive exotic aromatic forests are enough to
compute the values for general exotic aromatic forests in a straightforward
way.

The forests in the table are ordered according to their order as defined in
Definition 2.2 and by the number of roots.

Table 1: Butcher-Connes-Kreimer coproduct for all primitive exotic aromatic forests up to order three $\pi$ |  $\Delta_{BCK}(\pi)$
---|---
(b) |  $\mathbf{1}\otimes\forest{(b)}+\forest{(b)}\otimes\mathbf{1}$
b=b |  $\mathbf{1}\otimes\forest{b=b}+\forest{b=b}\otimes\mathbf{1}$
b |  $\mathbf{1}\otimes\forest{b}+\forest{b}\otimes\mathbf{1}$
1,1 |  $\mathbf{1}\otimes\forest{1,1}+\forest{1,1}\otimes\mathbf{1}$
(b[b]) |  $\mathbf{1}\otimes\forest{(b[b])}+\forest{b}\otimes\forest{(b)}+\forest{(b[b])}\otimes\mathbf{1}$
(b,b) |  $\mathbf{1}\otimes\forest{(b,b)}+\forest{(b,b)}\otimes\mathbf{1}$
(b[1,1]) |  $\mathbf{1}\otimes\forest{(b[1,1])}+\forest{1,1}\otimes\forest{(b)}+\forest{(b[1,1])}\otimes\mathbf{1}$
b=b[b] |  $\mathbf{1}\otimes\forest{b=b[b]}+\forest{b}\otimes\forest{b=b}+\forest{b=b[b]}\otimes\mathbf{1}$
b=b[1,1] |  $\mathbf{1}\otimes\forest{b=b[1,1]}+\forest{1,1}\otimes\forest{b=b}+\forest{b=b[1,1]}\otimes\mathbf{1}$
b[1]=b[1] |  $\mathbf{1}\otimes\forest{b[1]=b[1]}+\forest{1,1}\otimes\forest{b=b}+\forest{b[1]=b[1]}\otimes\mathbf{1}$
b[b] |  $\mathbf{1}\otimes\forest{b[b]}+\forest{b}\otimes\forest{b}+\forest{b[b]}\otimes\mathbf{1}$
b[1,1] |  $\mathbf{1}\otimes\forest{b[1,1]}+\forest{1,1}\otimes\forest{b}+\forest{b[1,1]}\otimes\mathbf{1}$
(b[1]),1 |  $\mathbf{1}\otimes\forest{(b[1]),1}+\forest{1,1}\otimes\forest{(b)}+\forest{(b[1]),1}\otimes\mathbf{1}$
b=b[1],1 |  $\mathbf{1}\otimes\forest{b=b[1],1}+\forest{1,1}\otimes\forest{b=b}+\forest{b=b[1],1}\otimes\mathbf{1}$
b[1],1 |  $\mathbf{1}\otimes\forest{b[1],1}+\forest{1,1}\otimes\forest{b}+\forest{b[1],1}\otimes\mathbf{1}$
(b[b[b]]) |  $\mathbf{1}\otimes\forest{(b[b[b]])}+\forest{b}\otimes\forest{(b[b])}+\forest{b[b]}\otimes\forest{(b)}+\forest{(b[b[b]])}\otimes\mathbf{1}$
(b[b,b]) |  $\mathbf{1}\otimes\forest{(b[b,b])}+2\forest{b}\otimes\forest{(b[b])}+\forest{b,b}\otimes\forest{(b)}+\forest{(b[b,b])}\otimes\mathbf{1}$
(b[b],b) |  $\mathbf{1}\otimes\forest{(b[b],b)}+\forest{b}\otimes\forest{(b,b)}+\forest{(b[b],b)}\otimes\mathbf{1}$
(b,b,b) |  $\mathbf{1}\otimes\forest{(b,b,b)}+\forest{(b,b,b)}\otimes\mathbf{1}$
(b[b[1,1]]) |  $\mathbf{1}\otimes\forest{(b[b[1,1]])}+\forest{1,1}\otimes\forest{(b[b])}+\forest{b[1,1]}\otimes\forest{(b)}+\forest{(b[b[1,1]])}\otimes\mathbf{1}$
(b[b[1],1]) |  $\mathbf{1}\otimes\forest{(b[b[1],1])}+\forest{1,1}\otimes\forest{(b[b])}+\forest{b[1],1}\otimes\forest{(b)}+\forest{(b[b[1],1])}\otimes\mathbf{1}$
(b[b,1,1]) |  $\mathbf{1}\otimes\forest{(b[b,1,1])}+\forest{1,1}\otimes\forest{(b[b])}+\forest{b}\otimes\forest{(b[1,1])}+\forest{b,1,1}\otimes\forest{(b)}+\forest{(b[b,1,1])}\otimes\mathbf{1}$
(b[1]),(b[1]) |  $\mathbf{1}\otimes\forest{(b[1]),(b[1])}+\forest{1,1}\otimes\forest{(b),(b)}+\forest{(b[1]),1}\otimes\forest{(b)}+\forest{(b[1]),1}\otimes\forest{(b)}+\forest{(b[1]),(b[1])}\otimes\mathbf{1}$
(b[1],b[1]) |  $\mathbf{1}\otimes\forest{(b[1],b[1])}+\forest{1,1}\otimes\forest{(b,b)}+\forest{(b[1],b[1])}\otimes\mathbf{1}$
(b[1,1],b) |  $\mathbf{1}\otimes\forest{(b[1,1],b)}+\forest{1,1}\otimes\forest{(b,b)}+\forest{(b[1,1],b)}\otimes\mathbf{1}$
(b[1,1,2,2]) |  $\mathbf{1}\otimes\forest{(b[1,1,2,2])}+2\forest{1,1}\otimes\forest{(b[1,1])}+\forest{1,1,2,2}\otimes\forest{(b)}+\forest{(b[1,1,2,2])}\otimes\mathbf{1}$
b=b[b[b]] |  $\mathbf{1}\otimes\forest{b=b[b[b]]}+\forest{b}\otimes\forest{b=b[b]}+\forest{b[b]}\otimes\forest{b=b}+\forest{b=b[b[b]]}\otimes\mathbf{1}$
b=b[b,b] |  $\mathbf{1}\otimes\forest{b=b[b,b]}+2\forest{b}\otimes\forest{b=b[b]}+\forest{b,b}\otimes\forest{b=b}+\forest{b=b[b,b]}\otimes\mathbf{1}$
b[b]=b[b] |  $\mathbf{1}\otimes\forest{b[b]=b[b]}+2\forest{b}\otimes\forest{b=b[b]}+\forest{b,b}\otimes\forest{b=b}+\forest{b[b]=b[b]}\otimes\mathbf{1}$
b=b[b[1,1]] |  $\mathbf{1}\otimes\forest{b=b[b[1,1]]}+\forest{1,1}\otimes\forest{b=b[b]}+\forest{b[1,1]}\otimes\forest{b=b}+\forest{b=b[b[1,1]]}\otimes\mathbf{1}$
b=b[b[1],1] |  $\mathbf{1}\otimes\forest{b=b[b[1],1]}+\forest{1,1}\otimes\forest{b=b[b]}+\forest{b[1],1}\otimes\forest{b=b}+\forest{b=b[b[1],1]}\otimes\mathbf{1}$
b[1]=b[b[1]] |  $\mathbf{1}\otimes\forest{b[1]=b[b[1]]}+\forest{1,1}\otimes\forest{b=b[b]}+\forest{b[1],1}\otimes\forest{b=b}+\forest{b[1]=b[b[1]]}\otimes\mathbf{1}$
b=b[b,1,1] |  $\mathbf{1}\otimes\forest{b=b[b,1,1]}+\forest{1,1}\otimes\forest{b=b[b]}+\forest{b}\otimes\forest{b=b[1,1]}+\forest{b,1,1}\otimes\forest{b=b}+\forest{b=b[b,1,1]}\otimes\mathbf{1}$
b[1]=b[b,1] |  $\mathbf{1}\otimes\forest{b[1]=b[b,1]}+\forest{1,1}\otimes\forest{b=b[b]}+\forest{b}\otimes\forest{b[1]=b[1]}+\forest{b,1,1}\otimes\forest{b=b}+\forest{b[1]=b[b,1]}\otimes\mathbf{1}$
b[1,1]=b[b] |  $\mathbf{1}\otimes\forest{b[1,1]=b[b]}+\forest{1,1}\otimes\forest{b=b[b]}+\forest{b}\otimes\forest{b[1,1]=b}+\forest{b,1,1}\otimes\forest{b=b}+\forest{b[1,1]=b[b]}\otimes\mathbf{1}$
b=b[1,1,2,2] |  $\mathbf{1}\otimes\forest{b=b[1,1,2,2]}+2\forest{1,1}\otimes\forest{b=b[1,1]}+\forest{1,1,2,2}\otimes\forest{b=b}+\forest{b=b[1,1,2,2]}\otimes\mathbf{1}$
b[2]=b[1,1,2] |  $\mathbf{1}\otimes\forest{b[2]=b[1,1,2]}+\forest{1,1}\otimes\forest{b=b[1,1]}+\forest{1,1}\otimes\forest{b[1]=b[1]}+\forest{1,1,2,2}\otimes\forest{b=b}+\forest{b[2]=b[1,1,2]}\otimes\mathbf{1}$
b[2,2]=b[1,1] |  $\mathbf{1}\otimes\forest{b[2,2]=b[1,1]}+2\forest{1,1}\otimes\forest{b=b[1,1]}+\forest{1,1,2,2}\otimes\forest{b=b}+\forest{b[2,2]=b[1,1]}\otimes\mathbf{1}$
b[1,2]=b[1,2] |  $\mathbf{1}\otimes\forest{b[1,2]=b[1,2]}+2\forest{1,1}\otimes\forest{b[1]=b[1]}+\forest{1,1,2,2}\otimes\forest{b=b}+\forest{b[1,2]=b[1,2]}\otimes\mathbf{1}$
b=b[1],(b[1]) |  $\mathbf{1}\otimes\forest{b=b[1],(b[1])}+\forest{1,1}\otimes\forest{b=b,(b)}+\forest{b=b[1],1}\otimes\forest{(b)}+\forest{(b[1]),1}\otimes\forest{b=b}+\forest{b=b[1],(b[1])}\otimes\mathbf{1}$
b=b[1],b[1]=b |  $\mathbf{1}\otimes\forest{b=b[1],b[1]=b}+\forest{1,1}\otimes\forest{b=b,b=b}+2\forest{b=b[1],1}\otimes\forest{b=b}+\forest{b=b[1],b[1]=b}\otimes\mathbf{1}$
b[b[b]] |  $\mathbf{1}\otimes\forest{b[b[b]]}+\forest{b}\otimes\forest{b[b]}+\forest{b[b]}\otimes\forest{b}+\forest{b[b[b]]}\otimes\mathbf{1}$
b[b,b] |  $\mathbf{1}\otimes\forest{b[b,b]}+2\forest{b}\otimes\forest{b[b]}+\forest{b,b}\otimes\forest{b}+\forest{b[b,b]}\otimes\mathbf{1}$
b[b[1,1]] |  $\mathbf{1}\otimes\forest{b[b[1,1]]}+\forest{1,1}\otimes\forest{b[b]}+\forest{b[1,1]}\otimes\forest{b}+\forest{b[b[1,1]]}\otimes\mathbf{1}$
b[b[1],1] |  $\mathbf{1}\otimes\forest{b[b[1],1]}+\forest{1,1}\otimes\forest{b[b]}+\forest{b[1],1}\otimes\forest{b}+\forest{b[b[1],1]}\otimes\mathbf{1}$
b[b,1,1] |  $\mathbf{1}\otimes\forest{b[b,1,1]}+\forest{1,1}\otimes\forest{b[b]}+\forest{b}\otimes\forest{b[1,1]}+\forest{b,1,1}\otimes\forest{b}+\forest{b[b,1,1]}\otimes\mathbf{1}$
b[1,1,2,2] |  $\mathbf{1}\otimes\forest{b[1,1,2,2]}+2\forest{1,1}\otimes\forest{b[1,1]}+\forest{1,1,2,2}\otimes\forest{b}+\forest{b[1,1,2,2]}\otimes\mathbf{1}$
(b[b[1]]),1 |  $\mathbf{1}\otimes\forest{(b[b[1]]),1}+\forest{1,1}\otimes\forest{(b[b])}+\forest{b[1],1}\otimes\forest{(b)}+\forest{(b[b[1]]),1}\otimes\mathbf{1}$
(b[b,1]),1 |  $\mathbf{1}\otimes\forest{(b[b,1]),1}+\forest{1,1}\otimes\forest{(b[b])}+\forest{b}\otimes\forest{(b[1]),1}+\forest{b,1,1}\otimes\forest{(b)}+\forest{(b[b,1]),1}\otimes\mathbf{1}$
(b[1]),b[1] |  $\mathbf{1}\otimes\forest{(b[1]),b[1]}+\forest{1,1}\otimes\forest{(b),b}+\forest{b[1],1}\otimes\forest{(b)}+\forest{(b[1]),1}\otimes\forest{b}+\forest{(b[1]),b[1]}\otimes\mathbf{1}$
(b,b[1]),1 |  $\mathbf{1}\otimes\forest{(b,b[1]),1}+\forest{1,1}\otimes\forest{(b,b)}+\forest{(b,b[1]),1}\otimes\mathbf{1}$
(b[1,2,2]),1 |  $\mathbf{1}\otimes\forest{(b[1,2,2]),1}+\forest{1,1}\otimes\forest{(b[1]),1}+\forest{1,1}\otimes\forest{(b[1,1])}+\forest{1,1,2,2}\otimes\forest{(b)}+\forest{(b[1,2,2]),1}\otimes\mathbf{1}$
b=b[b[1]],1 |  $\mathbf{1}\otimes\forest{b=b[b[1]],1}+\forest{1,1}\otimes\forest{b=b[b]}+\forest{b[1],1}\otimes\forest{b=b}+\forest{b=b[b[1]],1}\otimes\mathbf{1}$
b[b[1]],1 |  $\mathbf{1}\otimes\forest{b[b[1]],1}+\forest{1,1}\otimes\forest{b[b]}+\forest{b[1],1}\otimes\forest{b}+\forest{b[b[1]],1}\otimes\mathbf{1}$
b[b,1],1 |  $\mathbf{1}\otimes\forest{b[b,1],1}+\forest{1,1}\otimes\forest{b[b]}+\forest{b}\otimes\forest{b[1],1}+\forest{b,1,1}\otimes\forest{b}+\forest{b[b,1],1}\otimes\mathbf{1}$
b[1],b[1] |  $\mathbf{1}\forest{b[1],b[1]}+\forest{1,1}\otimes\forest{b,b}+2\forest{b[1],1}\otimes\forest{b}+\forest{b[1],b[1]}\otimes\mathbf{1}$
b[1,2,2],1 |  $\mathbf{1}\otimes\forest{b[1,2,2],1}+\forest{1,1}\otimes\forest{b[1],1}+\forest{1,1}\otimes\forest{b[1,1]}+\forest{1,1,2,2}\otimes\forest{b}+\forest{b[1,2,2],1}\otimes\mathbf{1}$
b=b[b,1],1 |  $\mathbf{1}\otimes\forest{b=b[b,1],1}+\forest{1,1}\otimes\forest{b=b[b]}+\forest{b}\otimes\forest{b=b[1],1}+\forest{b,1,1}\otimes\forest{b=b}+\forest{b=b[b,1],1}\otimes\mathbf{1}$
b[1]=b[b],1 |  $\mathbf{1}\otimes\forest{b[1]=b[b],1}+\forest{1,1}\otimes\forest{b=b[b]}+\forest{b}\otimes\forest{b[1]=b,1}+\forest{b,1,1}\otimes\forest{b=b}+\forest{b[1]=b[b],1}\otimes\mathbf{1}$
b=b[1,1,2],2 |  $\mathbf{1}\otimes\forest{b=b[1,1,2],2}+\forest{1,1}\otimes\forest{b=b[1,1]}+\forest{1,1}\otimes\forest{b=b[1],1}+\forest{1,1,2,2}\otimes\forest{b=b}+\forest{b=b[1,1,2],2}\otimes\mathbf{1}$
b[2]=b[1,1],2 |  $\mathbf{1}\otimes\forest{b[2]=b[1,1],2}+\forest{1,1}\otimes\forest{b=b[1,1]}+\forest{1,1}\otimes\forest{b[1]=b,1}+\forest{1,1,2,2}\otimes\forest{b=b}+\forest{b[2]=b[1,1],2}\otimes\mathbf{1}$
b[1]=b[1,2],2 |  $\mathbf{1}\otimes\forest{b[1]=b[1,2],2}+\forest{1,1}\otimes\forest{b[1]=b[1]}+\forest{1,1}\otimes\forest{b=b[1],1}+\forest{1,1,2,2}\otimes\forest{b=b}+\forest{b[1]=b[1,2],2}\otimes\mathbf{1}$
(b[1,2]),1,2 |  $\mathbf{1}\otimes\forest{(b[1,2]),1,2}+2\forest{1,1}\otimes\forest{(b[1]),1}+\forest{1,1,2,2}\otimes\forest{(b)}+\forest{(b[1,2]),1,2}\otimes\mathbf{1}$
b[1,2],1,2 |  $\mathbf{1}\otimes\forest{b[1,2],1,2}+2\forest{1,1}\otimes\forest{b[1],1}+\forest{1,1,2,2}\otimes\forest{b}+\forest{b[1,2],1,2}\otimes\mathbf{1}$
b[1]=b[2],1,2 |  $\mathbf{1}\otimes\forest{b[1]=b[2],1,2}+2\forest{1,1}\otimes\forest{b[1]=b,1}+\forest{1,1,2,2}\otimes\forest{b=b}+\forest{b[1]=b[2],1,2}\otimes\mathbf{1}$
b=b[1,2],1,2 |  $\mathbf{1}\otimes\forest{b=b[1,2],1,2}+2\forest{1,1}\otimes\forest{b=b[1],1}+\forest{1,1,2,2}\otimes\forest{b=b}+\forest{b=b[1,2],1,2}\otimes\mathbf{1}$

We refer to the discussion from Section 2.2 for an efficient way to compute
the CEM coaction. The computations can be checked using Proposition 4.17 with
the property

$\langle\Delta_{CEM}(\pi),\pi_{1}\otimes\pi_{2}\rangle=\frac{\sigma(\pi)\langle\pi_{1}\triangleright\pi_{2},\pi\rangle}{\sigma(\pi_{1})\sigma(\pi_{2})}.$

Table 2: CEM coaction for all exotic aromas and primitive exotic aromatic forests up to order three $\pi$ |  $\Delta_{CEM}(\pi)$
---|---
(b) |  $\forest{b}\otimes\forest{(b)}$
b=b |  $\forest{b,b}\otimes\forest{b=b}$
b |  $\forest{b}\otimes\forest{b}$
1,1 |  $\mathbf{1}\otimes\forest{1,1}$
(b[b]) |  $\forest{b,b}\otimes\forest{(b[b])}+\forest{(b),b}\otimes\forest{(b)}+\forest{b[b]}\otimes\forest{(b)}$
(b,b) |  $\forest{b,b}\otimes\forest{(b,b)}+2\forest{b[b]}\otimes\forest{(b)}$
(b[1,1]) |  $\forest{b}\otimes\forest{(b[1,1])}+2\forest{(b[1]),1}\otimes\forest{(b)}+\forest{b[1,1]}\otimes\forest{(b)}$
b=b[b] |  $\forest{b,b,b}\otimes\forest{b=b[b]}+\forest{b=b,b}\otimes\forest{(b)}+\forest{b,b[b]}\otimes\forest{b=b}$
b=b[1,1] |  $\forest{b,b}\otimes\forest{b=b[1,1]}+2\forest{b=b[1],1}\otimes\forest{(b)}+\forest{b,b[1,1]}\otimes\forest{b=b}$
b[1]=b[1] |  $\forest{b,b}\otimes\forest{b[1]=b[1]}+2\forest{b=b[1],1}\otimes\forest{(b)}$
b[b] |  $\forest{b,b}\otimes\forest{b[b]}+\forest{b[b]}\otimes\forest{b}$
b[1,1] |  $\forest{b}\otimes\forest{b[1,1]}+\forest{b[1,1]}\otimes\forest{b}$
(b),b |  $\forest{b,b}\otimes\forest{(b),b}+\forest{(b),b}\otimes\forest{b}$
(b[1]),1 |  $\forest{b}\otimes\forest{(b[1]),1}+\forest{(b[1]),1}\otimes\forest{b}$
b=b,b |  $\forest{b,b,b}\otimes\forest{b=b,b}+\forest{b=b,b}\otimes\forest{b}$
b=b[1],1 |  $\forest{b,b}\otimes\forest{b=b[1],1}+\forest{b=b[1],1}\otimes\forest{b}$
b[1],1 |  $\forest{b}\otimes\forest{b[1],1}$
(b),1,1 |  $\forest{b}\otimes\forest{(b),1,1}$
b=b,1,1 |  $\forest{b,b}\otimes\forest{b=b,1,1}$
(b[b[b]]) |  $\forest{b,b,b}\otimes\forest{(b[b[b]])}+2\forest{b[b],b}\otimes\forest{(b[b])}+\forest{(b),b[b]}\otimes\forest{(b)}+\forest{(b[b]),b}\otimes\forest{(b)}+\forest{b[b[b]]}\otimes\forest{(b)}$
(b[b,b]) |  $\forest{b,b,b}\otimes\forest{(b[b,b])}+2\forest{b[b],b}\otimes\forest{(b[b])}+2\forest{(b[b]),b}\otimes\forest{(b)}+\forest{b[b,b]}\otimes\forest{(b)}$
(b[b],b) |  $\forest{b,b,b}\otimes\forest{(b[b],b)}+\forest{b[b],b}\otimes\forest{(b,b)}+2\forest{b[b],b}\otimes\forest{(b[b])}+\forest{b[b,b]}\otimes\forest{(b)}+\forest{(b,b),b}\otimes\forest{(b)}$ $+\forest{b[b[b]]}\otimes\forest{(b)}$
(b,b,b) |  $\forest{b,b,b}\otimes\forest{(b,b,b)}+3\forest{b[b],b}\otimes\forest{(b,b)}+3\forest{b[b[b]]}\otimes\forest{(b)}$
(b[b[1,1]]) |  $\forest{b,b}\otimes\forest{(b[b[1,1]])}+\forest{b[1,1],b}\otimes\forest{(b[b])}+\forest{b[b]}\otimes\forest{(b[1,1])}+\forest{(b),b[1,1]}\otimes\forest{(b)}$ $+2\forest{(b[b[1]]),1}\otimes\forest{(b)}+\forest{b[b[1,1]]}\otimes\forest{(b)}$
(b[b[1],1]) |  $\forest{b,b}\otimes\forest{(b[b[1],1])}+\forest{b[b]}\otimes\forest{(b[1,1])}+\forest{(b[b[1]]),1}\otimes\forest{(b)}+\forest{(b[b,1]),1}\otimes\forest{(b)}$ $+\forest{b[b[1],1]}\otimes\forest{(b)}$
(b[b,1,1]) |  $\forest{b,b}\otimes\forest{(b[b,1,1])}+\forest{b[b]}\otimes\forest{(b[1,1])}+\forest{b[1,1],b}\otimes\forest{(b[b])}+\forest{(b[1,1]),b}\otimes\forest{(b)}$ $+2\forest{(b[b,1]),1}\otimes\forest{(b)}+\forest{b[b,1,1]}\otimes\forest{(b)}$
(b[1]),(b[1]) |  $\forest{b,b}\otimes\forest{(b[1]),(b[1])}+2\forest{(b),b}\otimes\forest{(b[1,1])}+2\forest{(b[1]),1,b}\otimes\forest{(b[b])}+2\forest{(b[1]),(b),1}\otimes\forest{(b)}$ $+2\forest{(b[1]),b[1]}\otimes\forest{(b)}$
(b[1],b[1]) |  $\forest{b,b}\otimes\forest{(b[1],b[1])}+2\forest{b[b[1],1]}\otimes\forest{(b)}+2\forest{(b[1],b),1}\otimes\forest{(b)}+2\forest{b[b]}\otimes\forest{(b[1,1])}$
(b[1,1],b) |  $\forest{b,b}\otimes\forest{(b[1,1],b)}+\forest{b[1,1],b}\otimes\forest{(b,b)}+\forest{b[b]}\otimes\forest{(b[1,1])}+\forest{b[b[1,1]]}\otimes\forest{(b)}$ $+2\forest{(b[1],b),1}\otimes\forest{(b)}+\forest{b[1,1,b]}\otimes\forest{(b)}$
(b[1,1,2,2]) |  $\forest{b}\otimes\forest{(b[1,1,2,2])}+2\forest{b[1,1]}\otimes\forest{(b[1,1])}+4\forest{(b[1,1,2]),2}\otimes\forest{(b)}+\forest{b[1,1,2,2]}\otimes\forest{(b)}$
b=b[b[b]] |  $\forest{b,b,b,b}\otimes\forest{b=b[b[b]]}+2\forest{b[b],b,b}\otimes\forest{b=b[b]}+\forest{b=b,b[b]}\otimes\forest{(b)}+\forest{b=b[b],b}\otimes\forest{(b)}$ $+\forest{b,b[b[b]]}\otimes\forest{b=b}$
b=b[b,b] |  $\forest{b,b,b,b}\otimes\forest{b=b[b,b]}+2\forest{b[b],b,b}\otimes\forest{b=b[b]}+2\forest{b=b[b],b}\otimes\forest{(b)}+\forest{b,b[b,b]}\otimes\forest{b=b}$
b[b]=b[b] |  $\forest{b,b,b,b}\otimes\forest{b[b]=b[b]}+\forest{b[b],b,b}\otimes\forest{b=b[b]}+2\forest{b=b[b],b}\otimes\forest{(b)}+\forest{b[b],b[b]}\otimes\forest{b=b}$
b=b[b[1,1]] |  $\forest{b,b,b}\otimes\forest{b=b[b[1,1]]}+\forest{b[1,1],b,b}\otimes\forest{b=b[b]}+\forest{b[b],b}\otimes\forest{b=b[1,1]}+\forest{b=b,b[1,1]}\otimes\forest{(b)}$ $+\forest{b=b[b[1]],1}\otimes\forest{(b)}+\forest{b,b[b[1,1]]}\otimes\forest{b=b}$
b=b[b[1],1] |  $\forest{b,b,b}\otimes\forest{b=b[b[1],1]}+\forest{b[b],b}\otimes\forest{b=b[1,1]}+\forest{b=b[b[1]],1}\otimes\forest{(b)}+\forest{b=b[b,1],1}\otimes\forest{(b)}$ $+\forest{b=b[1],b[1]}\otimes\forest{(b)}+\forest{b,b[b[1],1]}\otimes\forest{b=b}$
b[1]=b[b[1]] |  $\forest{b,b,b}\otimes\forest{b[1]=b[b[1]]}+\forest{b=b[b[1]],1}\otimes\forest{(b)}+\forest{b[1]=b[b],1}\otimes\forest{(b)}+\forest{b[1]=b,b[1]}\otimes\forest{(b)}$ $+\forest{b[b],b}\otimes\forest{b[1]=b[1]}$
b=b[b,1,1] |  $\forest{b,b,b}\otimes\forest{b=b[b,1,1]}+\forest{b,b[b]}\otimes\forest{b=b[1,1]}+\forest{b,b,b[1,1]}\otimes\forest{b=b[b]}+\forest{b=b[1],1,b}\otimes\forest{(b[b])}$ $+\forest{b=b[1,1],b}\otimes\forest{(b)}+2\forest{b=b[b,1],1}\otimes\forest{(b)}+\forest{b,b[b,1,1]}\otimes\forest{b=b}$
b[1]=b[b,1] |  $\forest{b,b,b}\otimes\forest{b[1]=b[b,1]}+\forest{b,b[b]}\otimes\forest{b[1]=b[1]}+\forest{b[1]=b,1,b}\otimes\forest{(b[b])}+\forest{b=b[b,1],1}\otimes\forest{(b)}$ $+\forest{b[1]=b[b],1}\otimes\forest{(b)}+\forest{b[1]=b[1],b}\otimes\forest{(b)}$
b[1,1]=b[b] |  $\forest{b,b,b}\otimes\forest{b[1,1]=b[b]}+\forest{b,b[b]}\otimes\forest{b[1,1]=b}+\forest{b[1,1],b,b}\otimes\forest{b=b[b]}+\forest{b[1]=b,1,b}\otimes\forest{(b[b])}$ $+\forest{b[1,1]=b,b}\otimes\forest{(b)}+2\forest{b[1]=b[b],1}\otimes\forest{(b)}+\forest{b[1,1],b[b]}\otimes\forest{b=b}$
b=b[1,1,2,2] |  $\forest{b,b}\otimes\forest{b=b[1,1,2,2]}+2\forest{b[1,1],b}\otimes\forest{b=b[1,1]}+4\forest{b=b[1,1,2],2}\otimes\forest{(b)}+\forest{b,b[1,1,2,2]}\otimes\forest{b=b}$
b[2]=b[1,1,2] |  $\forest{b,b}\otimes\forest{b[2]=b[1,1,2]}+\forest{b[1,1],b}\otimes\forest{b[1]=b[1]}+\forest{b=b[1,1,2],2}\otimes\forest{(b)}+2\forest{b[2]=b[1,2],1}\otimes\forest{(b)}$ $+\forest{b[2]=b[1,1],2}\otimes\forest{(b)}$
b[2,2]=b[1,1] |  $\forest{b,b}\otimes\forest{b[2,2]=b[1,1]}+2\forest{b[1,1],b}\otimes\forest{b=b[1,1]}+4\forest{b[2]=b[1,1],2}\otimes\forest{(b)}+\forest{b[1,1],b[2,2]}\otimes\forest{b=b}$
b[1,2]=b[1,2] |  $\forest{b,b}\otimes\forest{b[1,2]=b[1,2]}+4\forest{b[2]=b[1,1],2}\otimes\forest{(b)}$
b=b[1],(b[1]) |  $\forest{b,b,b}\otimes\forest{b=b[1],(b[1])}+\forest{b=b[1],1,b}\otimes\forest{(b[b])}+\forest{b,b,(b[1]),1}\otimes\forest{b=b[b]}+\forest{(b),b}\cdot\forest{b}\otimes\forest{b=b[1,1]}$ $+\forest{b=b,1,(b[1])}\otimes\forest{(b)}+\forest{(b),1,b=b[1]}\otimes\forest{(b)}+\forest{b,(b[1]),b[1]}\otimes\forest{b=b}$
b=b[1],b[1]=b |  $\forest{b,b,b,b}\otimes\forest{b=b[1],b[1]=b}+2\forest{b=b[1],1,b,b}\otimes\forest{b=b[b]}+2\forest{b=b,b=b[1],1}\otimes\forest{(b)}$
b[b[b]] |  $\forest{b,b,b}\otimes\forest{b[b[b]]}+2\forest{b[b],b}\otimes\forest{b[b]}+\forest{b[b[b]]}\otimes\forest{b}$
b[b,b] |  $\forest{b,b,b}\otimes\forest{b[b,b]}+2\forest{b[b],b}\otimes\forest{b[b]}+\forest{b[b,b]}\otimes\forest{b}$
b[b[1,1]] |  $\forest{b,b}\otimes\forest{b[b[1,1]]}+\forest{b[1,1],b}\otimes\forest{b[b]}+\forest{b[b]}\otimes\forest{b[1,1]}+\forest{b[b[1,1]]}\otimes\forest{b}$
b[b[1],1] |  $\forest{b,b}\otimes\forest{b[b[1],1]}+\forest{b[b]}\otimes\forest{b[1,1]}+\forest{b[b[1],1]}\otimes\forest{b}$
b[b,1,1] |  $\forest{b,b}\otimes\forest{b[b,1,1]}+\forest{b[1,1],b}\otimes\forest{b[b]}+\forest{b[b]}\otimes\forest{b[1,1]}+\forest{b[b,1,1]}\otimes\forest{b}$
b[1,1,2,2] |  $\forest{b}\otimes\forest{b[1,1,2,2]}+2\forest{b[1,1]}\otimes\forest{b[1,1]}+\forest{b[1,1,2,2]}\otimes\forest{b}$
(b),b[b] |  $\forest{b,b,b}\otimes\forest{(b),b[b]}+2\forest{(b),b}\cdot\forest{b}\otimes\forest{b[b]}+\forest{b[b]}\otimes\forest{(b),b}+\forest{(b),b[b]}\otimes\forest{b}$
(b[b]),b |  $\forest{b,b,b}\otimes\forest{(b[b]),b}+\forest{b[b],b}\otimes\forest{(b),b}+\forest{(b),b}\cdot\forest{b}\otimes\forest{(b),b}+\forest{(b[b]),b}\otimes\forest{b}$
(b,b),b |  $\forest{b,b,b}\otimes\forest{(b,b),b}+2\forest{b[b],b}\otimes\forest{(b),b}+\forest{(b,b),b}\otimes\forest{b}$
(b),b[1,1] |  $\forest{b,b}\otimes\forest{(b),b[1,1]}+\forest{(b),b}\otimes\forest{b[1,1]}+\forest{b[1,1]}\otimes\forest{(b),b}+\forest{(b),b[1,1]}\otimes\forest{b}$
(b),(b),b |  $\forest{b,b,b}\otimes\forest{(b),(b),b}+4\forest{(b),b}\cdot\forest{b}\otimes\forest{(b),b}+\forest{(b),(b),b}\otimes\forest{b}$
(b[1,1]),b |  $\forest{b,b}\otimes\forest{(b[1,1]),b}+\forest{b[1,1],b}\otimes\forest{(b),b}+2\forest{(b[1]),1,b}\otimes\forest{(b),b}+\forest{(b),b}\otimes\forest{b[1,1]}+\forest{(b[1,1]),b}\otimes\forest{b}$
(b),(b[1]),1 |  $\forest{b,b}\otimes\forest{(b),(b[1]),1}+\forest{(b),b}\otimes\forest{(b[1]),1}+\forest{b,(b[1]),1}\otimes\forest{(b),b}+\forest{(b),(b[1]),1}\otimes\forest{b}$
(b[b[1]]),1 |  $\forest{b,b}\otimes\forest{(b[b[1]]),1}+\forest{b[b]}\otimes\forest{(b[1]),1}+\forest{(b[b[1]]),1}\otimes\forest{b}$
(b[b,1]),1 |  $\forest{b,b}\otimes\forest{(b[b,1]),1}+\forest{b[b]}\otimes\forest{(b[1]),1}+\forest{(b[b,1]),1}\otimes\forest{b}$
(b[1]),b[1] |  $\forest{b,b}\otimes\forest{(b[1]),b[1]}+\forest{(b),b}\otimes\forest{b[1,1]}+\forest{(b[1]),1,b}\otimes\forest{b[b]}+\forest{(b[1]),b[1]}\otimes\forest{b}$
(b,b[1]),1 |  $\forest{b,b}\otimes\forest{(b,b[1]),1}+2\forest{b[b]}\otimes\forest{(b[1]),1}+\forest{(b,b[1]),1}\otimes\forest{b}$
(b[1,2,2]),1 |  $\forest{b}\otimes\forest{(b[1,2,2]),1}+\forest{b[1,1]}\otimes\forest{(b[1]),1}+\forest{(b[1]),1}\otimes\forest{b[1,1]}+\forest{(b[1,2,2]),1}\otimes\forest{b}$
b=b,b[b] |  $\forest{b,b,b,b}\otimes\forest{b=b,b[b]}+\forest{b,b,b[b]}\otimes\forest{b=b,b}+2\forest{b=b,b}\cdot\forest{b}\otimes\forest{b[b]}+\forest{b=b,b[b]}\otimes\forest{b}$
b=b[b],b |  $\forest{b,b,b,b}\otimes\forest{b=b[b],b}+\forest{b,b,b[b]}\otimes\forest{b=b,b}+\forest{b=b,b}\cdot\forest{b}\otimes\forest{b[b]}+\forest{b=b,b}\cdot\forest{b}\otimes\forest{(b),b}+\forest{b=b[b],b}\otimes\forest{b}$
b=b[1,1],b |  $\forest{b,b,b}\otimes\forest{b=b[1,1],b}+\forest{b,b,b[1,1]}\otimes\forest{b=b,b}+\forest{b=b,b}\otimes\forest{b[1,1]}+2\forest{b=b[1],1,b}\otimes\forest{(b),b}+\forest{b=b[1,1],b}\otimes\forest{b}$
b[1]=b[1],b |  $\forest{b,b,b}\otimes\forest{b[1]=b[1],b}+\forest{b=b,b}\otimes\forest{b[1,1]}+2\forest{b=b[1],1,b}\otimes\forest{(b),b}+\forest{b[1]=b[1],b}\otimes\forest{b}$
b=b,b[1,1] |  $\forest{b,b,b}\otimes\forest{b=b,b[1,1]}+\forest{b=b,b}\otimes\forest{b[1,1]}+\forest{b[1,1],b,b}\otimes\forest{b=b,b}+\forest{b=b,b[1,1]}\otimes\forest{b}$
(b),b=b,b |  $\forest{b,b,b,b}\otimes\forest{(b),b=b,b}+3\forest{(b),b}\cdot\forest{b,b}\otimes\forest{b=b,b}+2\forest{b=b,b}\cdot\forest{b,b}\otimes\forest{(b),b}+\forest{(b),b=b,b}\otimes\forest{b}$
b=b,b=b,b |  $\forest{b,b,b,b}\otimes\forest{b=b,b=b,b}+6\forest{b=b,b}\cdot\forest{b,b}\otimes\forest{b=b,b}+\forest{b=b,b=b,b}\otimes\forest{b}$
b=b,(b[1]),1 |  $\forest{b,b,b}\otimes\forest{b=b,(b[1]),1}+\forest{b=b,b}\otimes\forest{(b[1]),1}+\forest{b,b,(b[1]),1}\otimes\forest{b=b,b}+\forest{b=b,(b[1]),1}\otimes\forest{b}$
(b),b=b[1],1 |  $\forest{b,b,b}\otimes\forest{(b),b=b[1],1}+2\forest{(b),b}\cdot\forest{b}\otimes\forest{b=b[1],1}+\forest{b,b=b[1],1}\otimes\forest{(b),b}+\forest{(b),b=b[1],1}\otimes\forest{b}$
b=b,b=b[1],1 |  $\forest{b,b,b,b}\otimes\forest{b=b,b=b[1],1}+4\forest{b=b,b}\cdot\forest{b}\otimes\forest{b=b[1],1}+\forest{b=b[1],1,b,b}\otimes\forest{b=b,b}+\forest{b=b,b=b[1],1}\otimes\forest{b}$
b=b[b[1]],1 |  $\forest{b,b,b}\otimes\forest{b=b[b[1]],1}+\forest{b[b],b}\otimes\forest{b=b[1],1}+\forest{b=b[b[1]],1}\otimes\forest{b}$
b=b[b,1],1 |  $\forest{b,b,b}\otimes\forest{b=b[b,1],1}+\forest{b,b[b]}\otimes\forest{b=b[1],1}+\forest{b=b[1],1,b}\otimes\forest{b[b]}+\forest{b=b[b,1],1}\otimes\forest{b}$
b[1]=b[b],1 |  $\forest{b,b,b}\otimes\forest{b[1]=b[b],1}+\forest{b,b[b]}\otimes\forest{b[1]=b,1}+\forest{b[1]=b,1,b}\otimes\forest{b[b]}+\forest{b[1]=b[b],1}\otimes\forest{b}$
b=b[1,1,2],2 |  $\forest{b,b}\otimes\forest{b=b[1,1,2],2}+\forest{b[1,1],b}\otimes\forest{b=b[1],1}+\forest{b=b[1],1,b}\otimes\forest{b[1,1]}+\forest{b,b[1,1,2],2}\otimes\forest{b}$
b[2]=b[1,1],2 |  $\forest{b,b}\otimes\forest{b[2]=b[1,1],2}+\forest{b[1,1],b}\otimes\forest{b[1]=b,1}+\forest{b[2]=b,2}\otimes\forest{b[1,1]}+\forest{b[2]=b[1,1],2}\otimes\forest{b}$
b[1]=b[1,2],2 |  $\forest{b,b}\otimes\forest{b[1]=b[1,2],2}+\forest{b[1]=b[1,2],2}\otimes\forest{b}$
b[b[1]],1 |  $\forest{b,b}\otimes\forest{b[b[1]],1}+\forest{b[b]}\otimes\forest{b[1],1}$
b[b,1],1 |  $\forest{b,b}\otimes\forest{b[b,1],1}+\forest{b[b]}\otimes\forest{b[1],1}$
b[1],b[1] |  $\forest{b,b}\otimes\forest{b[1],b[1]}$
b[1,2,2],1 |  $\forest{b}\otimes\forest{b[1,2,2],1}+\forest{b[1,1]}\otimes\forest{b[1],1}$
(b),(b[1]),1 |  $\forest{b,b}\otimes\forest{(b),(b[1]),1}+\forest{(b),b}\otimes\forest{(b[1]),1}+\forest{b,(b[1]),1}\otimes\forest{(b),b}\forest{(b),(b[1]),1}\otimes\forest{b}$
(b[b]),1,1 |  $\forest{b,b}\otimes\forest{(b[b]),1,1}+\forest{(b),b}\otimes\forest{(b),1,1}$
(b,b),1,1 |  $\forest{b,b}\otimes\forest{(b,b),1,1}+2\forest{b[b]}\otimes\forest{(b),1,1}$
(b),(b),1,1 |  $\forest{b,b}\otimes\forest{(b),(b),1,1}+2\forest{(b),b}\otimes\forest{(b),1,1}$
(b[1,1]),2,2 |  $\forest{b}\otimes\forest{(b[1,1]),2,2}+\forest{b[1,1]}\otimes\forest{(b),1,1}+2\forest{(b[1]),1}\otimes\forest{(b),1,1}$
(b[1,2]),1,2 |  $\forest{b}\otimes\forest{(b[1,2]),1,2}+2\forest{(b[1]),1}\otimes\forest{b[1],1}$
b=b,b[1],1 |  $\forest{b,b,b}\otimes\forest{b=b,b[1],1}+\forest{b=b,b}\otimes\forest{b[1],1}$
b=b[b],1,1 |  $\forest{b,b,b}\otimes\forest{b=b[b],1,1}+\forest{b=b,b}\otimes\forest{(b),1,1}$
b=b[1,1],2,2 |  $\forest{b,b}\otimes\forest{b=b[1,1],2,2}+2\forest{b=b[1],1}\otimes\forest{(b),1,1}$
b[1]=b[1],2,2 |  $\forest{b,b}\otimes\forest{b[1]=b[1],2,2}+2\forest{b=b[1],1}\otimes\forest{(b),1,1}$
(b),b=b,1,1 |  $\forest{b,b,b}\otimes\forest{(b),b=b,1,1}+2\forest{(b),b}\cdot\forest{b}\otimes\forest{b=b,1,1}+\forest{b=b,b}\otimes\forest{(b),1,1}$
b=b,b=b,1,1 |  $\forest{b,b,b,b}\otimes\forest{b=b,b=b,1,1}+4\forest{b=b,b}\cdot\forest{b}\otimes\forest{b=b,1,1}$
b[1]=b[2],1,2 |  $\forest{b,b}\otimes\forest{b[1]=b[2],1,2}+2\forest{b[1]=b,1}\otimes\forest{b[1],1}$
b=b[1,2],1,2 |  $\forest{b,b}\otimes\forest{b=b[1,2],1,2}+2\forest{b[1]=b,1}\otimes\forest{b[1],1}$
b[1,2],1,2 |  $\forest{b}\otimes\forest{b[1,2],1,2}$