# A Study on the Reliability of Automatic Dysarthric Speech Assessments

Xavier F. Cadet, Ranya Aloufi, Sara Ahmadi-Abhari, Hamed Haddadi
Imperial College London Corresponding author<EMAIL_ADDRESS>
###### Abstract

Automating dysarthria assessments offers the opportunity to develop effective,
low-cost tools that address the current limitations of manual and subjective
assessments. Nonetheless, it is unclear whether current approaches rely on
dysarthria-related speech patterns or external factors. We aim toward
obtaining a clearer understanding of dysarthria patterns. To this extent, we
study the effects of noise in recordings, both through addition and reduction.
We design and implement a new method for visualizing and comparing feature
extractors and models, at a patient level, in a more interpretable way. We use
the UA-Speech dataset with a speaker-based split of the dataset. Results
reported in the literature appear to have been done irrespective of such
split, leading to models that may be overconfident due to data-leakage. We
hope that these results raise awareness in the research community regarding
the requirements for establishing reliable automatic dysarthria assessment
systems.

Index Terms: dysarthric speech, speech recognition

##  I Introduction

Dysarthria is caused by a lack of articulatory control and muscle weakness,
which affect the rate of speech, the dynamic amplitudes and pitches, and the
manner in which the spoken word is pronounced. All of which contribute to
unintelligible speech and difficulty in understanding, due to the inaccurate
articulation of phonemes and abnormal speech patterns [1].

Dysarthria classification has become increasingly important in diagnosing the
disorder, determining the best treatment options, and conducting speech
therapy sessions as needed [2, 3, 4, 5, 6, 7]. While the literature indicates
that existing automated assessment attempts achieve high accuracy, the
methodology by which the assessment models were evaluated may not be
sufficiently generalized, since the models have been trained and evaluated on
the same speakers or a single unseen speaker per intelligibility class [8].
There is a limited research examining the effectiveness of such assessments
under various scenarios, and provide fine-grained information about their
performance for individual patients.

Our main contribution in this paper is the development of an interpretable
tool that enables the aggregation and visualization of assessment results. The
tool has been used to empirically evaluate a number of representations (e.g.,
acoustics and self-supervised methods) and classifiers (e.g., logistics
regression and multilayer perceptron), as well as to carry out binary and
multiclass tasks (e.g., disease, word, and severity classification). In order
to simulate the real-world recordings collection scenarios, we experimented
with three settings: default, noise addition, and noise reduction. Our tool
can provide a better understanding of types of impairments that can lead to
empirical features, usable to develop programs that facilitate identification
of disorders and their characteristics. As an example, the severity
classification task performance under these three scenarios, and using
acoustics and HuBERT [9] features, shows that HuBERT outperforms the acoustics
algorithm.

Figure 1: The proposed tool overview.

##  II Dysarthria Automatic Assessments

Dysarthria intelligibility assessment is typically performed in two stages
[10]. The training stage involves building a computational model based on the
patients' speech samples and their respective speech intelligibility classes.
Once the model is trained, it can be used to identify classes of speakers with
unknown intelligibility levels, by comparing their acoustic features with
those which were used during training. Intelligibility assessment approaches
that are reference-free focus on developing classification models without any
prior understanding of healthy speech, instead focusing on the extraction of
acoustic features believed to be highly correlated with intelligibility [11].
A reference-based approach uses healthy speech signals as a standard for
measuring intelligibility [12]. Healthy speech data is utilized in these
approaches (e.g., ASR-based approaches) in order to determine the
characteristics of intelligible speech, and then, they are used as a basis for
estimating the level of intelligibility [13, 14]. It exploits the fact that
ASR systems trained only on healthy speech perform poorly on dysarthric
speech, and that the performance of ASR systems deteriorates with the severity
of dysarthric speech.

##  III Towards an Effective Dysarthria Assessments

TABLE I: Task Performance: Results are reported using Balanced Accuracy, across the same set of patients. For the Disease and Words classification problems, results are based on individual recordings. For the Intelligibility task, results are calculated performance at the speaker level.  Task | Extractor | Default | Noise Reduction | Noise Addition
---|---|---|---|---
|  | LR | MLP | LR | MLP | LR | MLP
disease | acoustic | 0.78 | 0.57 | 0.72 | 0.52 | 0.83 | 0.86
disease | wav2vec2 | 0.98 | 0.96 | 0.9 | 0.9 | 1.0 | 1.0
disease | hubert | 0.97 | 0.96 | 0.93 | 0.91 | 1.0 | 0.99
disease | modified cpc | 0.97 | 0.97 | 0.84 | 0.88 | 1.0 | 1.0
words | acoustic | 0.08 | 0.1 | 0.07 | 0.11 | 0.08 | 0.09
words | wav2vec2 | 0.59 | 0.63 | 0.45 | 0.48 | 0.56 | 0.59
words | hubert | 0.76 | 0.75 | 0.6 | 0.58 | 0.77 | 0.75
words | modified cpc | 0.61 | 0.66 | 0.51 | 0.55 | 0.62 | 0.66
intelligibility | acoustic | 0.2 | 0.5 | 0.2 | 0.2 | 0.2 | 0.2
intelligibility | wav2vec2 | 0.6 | 0.6 | 0.6 | 0.5 | 0.6 | 0.6
intelligibility | hubert | 0.6 | 0.6 | 0.5 | 0.6 | 0.6 | 0.6
intelligibility | modified cpc | 0.3 | 0.3 | 0.6 | 0.25 | 0.5 | 0.5

###  III-A Overview

Due to the fact that the effectiveness of dysarthria assessments is influenced
by the quality of recordings, the representations used, and the classification
algorithm used, we intend to develop an interpretable tool that will
facilitate the understanding of the outputs of such assessments. An overview
of the proposed tool is provided in Figure 1, which can be easily adapted to
extract a variety of features, followed by multiple classification algorithms.
Then, our tool aggregates the results per patient in order to verify the
reliability of the assessment results. Aggregation outputs could be
interpreted as intelligibility classes, such as low, mid, and high levels, and
could provide clinicians with an interpretable classification of the speaker's
intelligibility.

###  III-B Experimental Settings

Dataset. The UA-Speech [15] dataset contains recordings of control and
dysarthric speech. Speech signals are sampled at $16\text{\,}\mathrm{kHz}$ .
It comprises 13 healthy control speakers and 15 dysarthric speakers. The
vocabulary includes 455 distinct words with 10 digits, 26 radio alphabets, 19
computer commands, 100 common words and 300 uncommon words. Speakers were
divided into four different categories based on the severity of the condition,
namely high (H), mid (M), low (L) and very low (VL). In our experiments, we do
not consider the uncommon words.
Feature Extraction. We use a variety of handcrafted and learned features
reported in the literature.

  * •

_Hand-crafted Features._ Acoustic measures of articulation, voice, and prosody
were extracted using PRAAT [16, 17]. Examples include mean harmonic-to-noise
ratio (HNR), fraction of locally unvoiced frames, number of voice breaks,
degree of voice breaks, mean and standard deviation of pitch, jitter and
shimmer, and cepstral peak prominence (CPP) [18].

  * •

_Self-supervised Features._ Self-supervised learning (SSL) methods such as
wav2vec2 [19], CPC, and HuBERT [9] have been successfully exploited for a
variety of speech classification tasks [19, 20, 21]. Self-Supervised Feature
extractions are done using pre-trained models implemented by SUPERB [22]
benchmark111https://github.com/s3prl/s3prl, using a NVIDIA Quadro GV100 GPU.

Classification Tasks.

  * •

_Dysarthria Classification._ Dysarthria may be classified as flaccid, spastic,
ataxic, hypokinetic, choreatic, dystonic, or mixed. We treat this task as a
binary classification task, since there is no information regarding the type
of disease in the used dataset. The 13 control speakers belong to one class,
while the 15 dysarthric speakers belong to another class. We test both binary
and multi classification tasks as below. We compare performance of Logistic
Regression (LR) and Multi Layer Perceptron (MLP) Classifiers using Scikit-
learn version 1.2.1 [23]. We consider the default parameters for each model.
And operate patient-based splits such that 19 patients are used for training
and 9 for testing. We consider the following tasks:
newline

  * •

_Word Classification._ Dysarthria patients are more likely to be able to utter
isolated words rather than continuous sentences [12]. Isolated word
recognition is the process of converting the input speech command into the
corresponding text format [24]. Keyword spotting involves detecting specific
words or phrases within larger spoken sentences or utterances [25]. We
consider this task as multi classification task, with $155$ individual words.
We considered only words that were not identified as uncommon in the original
dataset.

  * •

_Severity Classification._ Dysarthria can vary in severity, leading to speech
of different degrees of intelligibility [4]. We consider $5$ classes, the UA-
Speech dataset provides $4$ severity levels, we add one for control speakers.
We proceed stratified splitting over the classes.

Metrics. We report the balanced accuracy to account for the classes' imbalance
present in the dataset across the different tasks [26]. For the Dysarthria and
word classification, performance is evaluated at the recording level. For
instance, we consider an audio sample to be classified as Dysarthric or non
Dysarthric speech. For the Severity classification, we report the performance
at a speaker level. We first train a model to predict the severity of the
different samples, then we group the predictions per patient. A patient
severity level is then based on the most frequent class predicted.

Figure 2: Speaker Intelligibility: The first and second rows are associated
with fine-grained intelligibility predictions, and use the same subset of
patients. All predictions are made using logistic regression-based
classifiers. The top row shows the predictions made using acoustic features
while the bottom shows uses HuBERT extracted features. From left to right, the
columns are Default, Noise Reduction, Noise Addition datasets.

###  III-C Analysis Configurations

####  III-C1 _Default Setting_

Q1.  _How reliable are classifiers when confronted to unknown speakers?_
Preliminaries. Classification tasks can help healthcare professionals to
better understand and manage the speech difficulties associated with
dysarthria, and to develop appropriate treatment plans to improve
communication. Although the literature provides references to existing
automated assessment attempts that have achieved high levels of accuracy, they
are not necessarily indicative of generalizability due to their evaluation
methodology [8, 10]. The training and evaluation of the models were conducted
on the same speakers or on only one unseen speaker for each target class [27].
Consequently, such results might be biased by the model ability to identify
the correct speaker rather than identifying dysarthria related information.
Setup. The raw recordings from the UA-Speech dataset are divided by patient ID
to ensure that the training and testing sets do not overlap. Thus, all the
recording from a speaker are either in the training or the test set. We
evaluate the performance of the classifiers when confronted to speakers that
were not used during training. We test the performance based on combinations
of features extractors and classification models. For the feature extraction,
we consider hand-crafted features through acoustic features and different
representations extracted by state-of-the-art self-supervised models such as
HuBERT and Wav2vec2, and CPC.
Results. Huang et al., in [8] shows that most existing work had achieved very
high accuracy ($75.1\%$ to $93.97\%$), however, most of these studies
validated their models using the same samples of speech used in training,
making their results less generalizable. Furthermore, based on the task at
hand, the target can be heavily imbalanced. Therefore, we opt for the Balanced
Accuracy Metric [26]. From table I, we observe that models trained using Self-
Supervised representations outperform models trained on acoustic features
across all tasks. Since the results are obtained without fine-tuning for the
self-supervised feature extractors, they appear as promising direction for
automated assessment of dysarthria. To get a better understanding of the
reliability of the assessment at the patient level, we use the proposed tool
in Section III-A, as show in figure 2, that allows for a fine-grained analysis
of the predictions of the models and feature extractors. The tool can be
adapted to other features extractors and classification models. Therefore,
such tool could provide clinicians with more interpretable assessment for a
given patient, and be used toward more personalized treatments.

Figure 3: Analysis configurations: default, noise addition, and noise
reduction.  Figure 4: Word Classification: The first and second rows are
associated with word-level accuracy, and use the same subset of patients. All
predictions are made using logistic regression-based classifiers. The top row
shows the predictions made using acoustic features while the bottom shows uses
HuBERT extracted features. From left to right, the columns are Default, Noise
Reduction, Noise Addition datasets.

####  III-C2 Noise Addition

Q2.  _Are models trained on the Default dataset biased by patterns related to
the noise in recordings?_
Preliminaries. Some recordings in the default settings of the dataset have
different noise level and clicking sounds. Thus, one could argue that the good
performance observed on the disease classification task is due to the presence
of patterns associated with external factors from the recording rather than
speech related information. To confirm whether the feature extractors and
models ability to leverage information that would be specific to classes, we
conduct audio sample mixing to exacerbate such effect.
Setup. To assert whether the performance observed are due to models learning
to distinguish based on specific noise pattern, we generate a new version of
the dataset. First, we obtained a single background noise sample from the WHAM
dataset [28]. Then, we mixed every audio recording from control patient with
that noise pattern. Under such scenario: features extractors and models able
to extract that singular noise pattern would achieve higher performance.
Results. From Table I, we observe that all combinations of feature extractor
and models have achieved higher balanced accuracy on the disease
classification tasks. Therefore, when enforcing a singular pattern across
control patients, all feature extractors and models were able to leverage such
pattern. Therefore, we conduct further analysis after reducing the noise
levels in the recordings.

####  III-C3 Noise Reduction

Q3.  _What impact does enhancing the recordings have over the different
tasks?_
Preliminaries. Given the observation from the Noise Addition setting, we
considered a scenario under which we enhance the Default dataset. One such way
is to use speech restoration methods. Speech restoration process involves
restoring degraded speech signals to their original quality [29]. For
instance, speech is typically surrounded by background noise, blurred by
reverberation in the room, or recorded with low-quality equipment. It is
possible that ambient noise from clinical clicks or other artifact potentially
present in Dysarthric recordings. In [30], the effectiveness of automatic
denoising tools is quite limited, particularly for dysarthric speakers with
severe grades of disorder.
Setup. Our objective was to enhance the recordings by applying one of the
speech enhancement approaches, and to evaluate the performance of the models
in such scenarios. We generate a new version of dataset after applying
`VoiceFixer` [31], a method that attempts to remove multiple distortions
simultaneously. The pre-trained `VoiceFixer` models produce outputs with
$44.1\text{\,}\mathrm{kHz}$ sample rate, therefore, we further apply
resampling to $16\text{\,}\mathrm{kHz}$ before extracting the representations.
This ensures that the sampling rates match between the input signal and the
feature extractor models pre-trained on data with $16\text{\,}\mathrm{kHz}$
sample rate.
Results. From table I, we observe that across all the tasks, feature
extractors and classifiers tested: the performance decreased compared to the
default setting. This indicates that performance observed in the default
settings might be partly due to patterns that are corrected post-enhancement.
We further investigate the variation at the patient level through our proposed
visualization tool of the models predictions for the speaker intelligibility
task. Figure 2 indicates major differences when using acoustic features.
Indeed, across all patients, the logistic regression models consistently
classified recording to the level of control recordings. Nonetheless, the
application of the enhancement tool lead to loss of information in some
recording: such as partial speech segments removal in recordings. This
indicates that systematic use of such enhancement tools requires additional
care, if used as a preprocessing step in automated assessment pipelines.

##  IV Conclusions and Future Work

Data Leakage on Medical Recordings. Depending on the dataset split operated,
results can be biased by data-leakage. Models that are trained and evaluated
on patients with recordings that are in both the training and evaluation set
might have high performance, but unreliable as they have been partly trained
on speaker specific properties. Ensuring dataset splits such that patients are
in exclusively in one set or another allows for investigating performance on
future, and therefore unknown at train time, patients.
Classes Imbalance. For the Intelligibility Severity assessment, there is a
major class imbalance with respect to the number of recordings, i.e., $8.3\%$
(VL), $14\%$ (L), $13.9\%$ (M), $63.8\%$ (H + C) or $19.6\%$ (H) and $44.2\%$
(C). This leads to classification models that are likely to struggle
distinguishing between the VL, L and M classes. Furthermore, the coarse labels
given to the intelligibility, namely, patients with $26\%$ intelligibility
score being grouped with patients with intelligibility scores of $49\%$, make
it difficult to provide tools that obtain a detailed understanding of these
patterns.
Limitations and Future Work. Future work includes the extension to other
datasets, and data augmentation methods to tackle classes imbalanced. In [27],
Nemours is a very small database as compared to other databases though it
focused only on spastic dysarthria. According to the severity levels, the
classification accuracy of their experiments ranges from $40.41\%$ to
$95.80\%$ using only acoustic features. We considered the balanced accuracy to
summarize performance of the models, nonetheless, class specific performance
analysis could be beneficial. We will open-source the evaluation toolkit. We
welcome the community to participate and drive the research frontier.

##  V Acknowledgements

The authors would like to thank Sandra Siby for numerous comments, technical
questions, references, and invaluable suggestions for the presentation that
led to an improved text. Xavier F. Cadet is supported by UK Research and
Innovation (UKRI Centre for Doctoral Training in AI for Healthcare grant
number EP/S023283/1). Hamed Haddadi is supported by the EPSRC Open Plus
Fellowship (EP/W005271/1: Securing the Next Billion Consumer Devices on the
Edge). For the purpose of open access, the authors have applied a Creative
Commons Attribution (CC BY) license to any Author Accepted Manuscript version
arising.

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# Generalized metric tree arrangements and Dressians

Ayush Kumar Tewari

###### Abstract

Metric trees and metric tree arrangements index cones in the polyhedral fan
structure in Dressian $\operatorname{\rm Dr}(2,n)$ and $\operatorname{\rm
Dr}(3,n)$ respectively. We study the adjacency of cones in Dressian
$\operatorname{\rm Dr}(3,n)$ and introduce general Whitehead moves which
provide a condition for adjacency of cones in $\operatorname{\rm Dr}(3,n)$ in
terms of the associated metric tree arrangements. Based on these results we
introduce the notion of generalized metric tree arrangements which
parameterize points in $\operatorname{\rm Dr}(k,n)$ and extend previously
known results to $\operatorname{\rm Dr}(k,n)$.

###### Contents

  1. 1 Introduction
  2. 2 Dressian
  3. 3 Adjacency of cones in Dr(3,n)
    1. 3.1 Adjacency in cones in Dr(3,n)
  4. 4 Generalized metric tree arrangements for Dr(k,n)
  5. A Appendix
    1. A.1 Metric tree arrangements for Dr(3,6)
    2. A.2 Adjacent cone tree arrangements in Dr(3,6)

##  1 Introduction

Tropical linear spaces are parameterized by the tropical prevariety Dressian
$Dr(k,n)$. The Dressian contains the tropicalization of the Grassmannian,
which is the tropical Grassmannian $\text{Trop}(Gr(k,n))$. As tropical
varieties can be understood as metric graphs, this perspective helps us
understand the space $\text{Trop}(Gr(k,n))$ and the associated tropical
prevariety i.e, Dressian $Dr(k,n)$ in the light of metric trees.

In [12] it is shown that $Dr(2,n)=\text{Trop}(Gr(2,n))$ is a space which is
equal to the _moduli space of phylogenetic trees_. Additionally, in [7] the
authors establish that the space of tropical planes $\text{Trop}(Gr(3,n))$ can
be understood in terms of _metric tree arrangement_. They compute explicitly
$Dr(3,6)$ and $Dr(3,7)$ and describe all maximal cones, up to symmetry, and
specify all the trivalent metric trees in the metric tree arrangement
associated to each maximal cone.

In this paper, we have two main objectives. Firstly, we try to understand the
adjacency conditions for maximal cones in $Dr(3,n)$. It is already known that
metric trees indexing adjacent maximal cones in $Dr(2,n)$ are related by a
_Whitehead move_ , which also correspond to _bistellar flips_ [4], which
perform a combinatorial restructuring between triangulations. Building on the
computations done on [7] $Dr(3,6)$, we consider adjacent cones in $Dr(3,6)$
and introduce the notion of a _generalized Whitehead move_ in Definition 13.
We show that two adjacent cones in $Dr(3,n)$ are related by a generalized
Whitehead move in Theorem 14. We formalize this result for $Dr(3,n)$ in the
form of Lemma 12.

Secondly, we try to explore the connection between metric tree arrangements
and $Dr(k,n)$. Prior to this work, this is only known to exist for $Dr(2,n)$
and $Dr(3,n)$. We introduce the notion of _generalized metric tree
arrangements_ and show that cones in $Dr(k,n)$ index a collection of metric
trees and also a generalized metric tree arrangement, and these two notions
coincide in the case of $Dr(3,n)$. This establishes the fact that our findings
can be considered as a genuine extension of the results in [7] as illustrated
by Corollary 23. We believe this would open new avenues to study higher order
Dressians, especially as we note that computing Dressians is computationally
expensive [9], [2] and we stress on the fact that computations for $Dr(4,8)$
are beyond the reach of current computational techniques and would require new
techniques, although we do hope that with our work, for example the
computations in Example 16; could provide some insights in developing these
new techniques.

The structure of the paper is as follows. Section 2 provides a brief
introduction to Dressian. In Section 3 we delve into the question of adjacency
for cones in $Dr(3,n)$, staring with adjacent cones in $Dr(3,n)$ and proving
the main technical result in Lemma 12, based on which we state Theorem 14,
which is our final takeaway in terms of understanding adjacency in $Dr(3,n)$.
Section 4 introduces the definition of generalized metric tree arrangements
and we show that these tree arrangements induce weights which reside in
$Dr(k,n)$ in Theorem 19. We establish equivalence of generalised metric tree
arrangements and regular matroidal subdivisions of $\Delta(k,n)$, which
connects our findings as the proper generalization of results concerning
$Dr(3,n)$ in [7]. Lastly, we have an appendix to provide descriptions of cones
and their associated metric tree arrangements for $Dr(3,6)$, based on which we
formulate our results or $Dr(3,n)$.

##  2 Dressian

We recall some basic definitions of Grassmanian, tropical Grassmannian and
Dressian [10]. The Grassmannian $Gr(d,n)$ is a smooth projective variety
defined by the Plücker ideal $I_{d,n}$,

$I_{d,n}=\langle\mathcal{P}_{I,J}:I,J\subseteq[n],|I|=d-1,|J|=d+1\rangle$

whose generators are the quadratic Plücker relations,

$\mathcal{P}_{I,J}=\sum_{j\in J}\text{sgn}(j;I,J)\cdot p_{I\cup j}\cdot
p_{J\setminus j}$

The Grassmanian parameterizes $(r-1)$ dimensional subspaces of
$\mathbb{P}^{n-1}$. The _tropical Grassmannian_ $\text{Trop}(Gr(d,n))$ is the
tropical variety defined by the Plücker ideal $I_{d,n}$. Among the generators
of the ideal $I_{d,n}$ are the _three term Plücker_ relations which are of the
form

$p_{Aij}p_{Akl}-p_{Aik}p_{Ajl}+p_{Ail}p_{Ajk}$

where $A\in$ $[n]\choose k-2$ and $i,j,k,l\in[n]\setminus A$ are pairwise
distinct. The three term Plücker relations generate the image of the ideal
$I_{d,n}$ in the Laurent polynomial ring $\mathbb{K}[p_{A}^{\pm 1}]$, where
$\mathbb{K}$ is a field and $A\in$ $[n]\choose k-2$. The _Dressian_
$\operatorname{\rm Dr}(d,n)$ is the tropical prevariety defined by all the
three term Plücker relations. The Grassmannian and Dressian have a pure
rational polyhedral fan structure in $\mathbb{R}^{{n\choose
d}-1}\cong\mathbb{R}^{n\choose d}/\mathbb{R}1$.

######  Remark 1.

It is mentioned in [11, Remark 12] that for the Dressian, considering all
three term relations is equivalent to considering all quadratic relations and
these two notions appear as strong and weak matroids over tropical hyperfields
defined in [1], which are both equivalent.

The Plücker relations define a natural Plücker fan structure on the Dressian
$\operatorname{\rm Dr}(d,n)$ where two weight vectors $\mu$ and $\mu^{\prime}$
are in the same cone if they specify the same initial form for each relation.
The Dressian $\operatorname{\rm Dr}(d,n)$ is also endowed with a secondary fan
structure as follows [13],

######  Theorem 2.

A vector $w\in\mathbb{R}^{n\choose d}$ lies in the Dressian $\operatorname{\rm
Dr}(d,n)$ if and only if it induces a matroidal subdivision of the
hypersimplex $\Delta(d,n)$.

It was shown in [11] that these two fan structures coincide.

######  Theorem 3.

The Plücker fan structure on the Dressian $\operatorname{\rm Dr}(d,n)$ as a
fan in $\mathbb{R}^{{n\choose d}-1}$ coincides with the secondary fan
structure.

This result is true in much more generality but for the purpose for our
discussion, Theorem 3 suffices.

We firstly discuss the case of $Dr(2,n)$. $\text{Trop}(Gr(2,n))$ enjoys a
strong connection with the space of phylogenetic trees which we recall here.

A _phylogenetic_ tree $T$ is a tree with $n$ labeled leaves and no vertices of
degree two. A phylogenetic tree $T$ is trivalent if for each vertex $v\in
V(T)$ either $\deg(v)=3$ or $\deg(v)=1$. To each such phylogenetic tree we can
attach a metric data in the following way. We assign a length
$l_{e}\in\mathbb{R}$ to each edge of $T$. Between any two leaves $i$ and $j$
of $T$, there is a unique path in $T$, and the distance between the leaves can
be obtained as sum of $l_{e}$ along this path. This gives us the _tree
distance_ vector $\delta=(\delta_{ij})\in\mathbb{R}^{n\choose 2}$. When all
$l_{e}$ are nonnegative, the tree distance gives a finite metric. Finite
metric spaces that arise from such metric tree are called _tree metric_. The
set of all tree distances is known as the space of phylogenetic trees on $n$
leaves and is denoted as $\mathcal{T}_{n}$ [10].

######  Theorem 4 (Theorem 4.3.5 [10]).

$Gr(2,n)=\text{Trop}(Gr(2,n))=Dr(2,n)=\mathcal{T}_{n}$

The space of metric trees naturally forms a polyhedral fan. The maximal cones
of this fan correspond to trivalent trees. Two labelled trivalent trees
$T_{1}$ and $T_{2}$ correspond to adjacent maximal cones if and only if they
differ by a Whitehead move. In Figure 1, a Whitehead move is given by choosing
an internal edge and modifying the tree as shown in the diagram. Note that the
Whitehead move involves a choice since there are two possible ways to modify
the trivalent tree.

Figure 1: Whitehead moves between three trivalent trees. The central figure
shows the tree corresponding to the cone which is the intersection of the
surrounding cones.

We recall definitions of hypersimplex, splits and associated results [8]. The
hypersimplex $\Delta(k,n)$ is the convex polytope obtained by intersecting the
unit cube $[0,1]^{n}$ with the hyperplane $\sum x_{i}=d$. Equivalently, it can
be defined as a $0-1$ polytope generated by the convex hull of all points with
$k$ ones and $n-k$ zeros. The facet of $\Delta(k,n)$ defined by the hyperplane
$x_{i}=0$ is referred as the _i-th deletion facet_ and the facet defined by
the hyperplane $x_{i}=1$ is referred as the _i-th contraction facet_ [7].

An important observation concerning deletion and contraction facets of
$\Delta(k,n)$ is the following [7],

######  Lemma 5.

Each deletion facet of $\Delta(k,n)$ is isomorphic to $\Delta(k,n-1)$ and each
contraction facet of $\Delta(k,n)$ is isomorphic to $\Delta(d-1,n-1)$.

######  Definition 6.

A split $S$ of a polytope $P$ is the subdivision with exactly two maximal
cells.

The hyperplane defining the split $S$ is called the _split hyperplane_. A
split is a regular subdivision [8, Lemma 3.5]. Two splits are said to be
_compatible_ if the corresponding split hyperplanes do not meet in the
interior of the polytope.

Let $\Delta(k,n)$ be the hypersimplex. The $(A,B;\mu)$-hyperplane [8] is
defined as follows. Let $A,B\subset[n]$ be non-empty sets such that $A\cup
B=[n]$ and $A\cap B=\emptyset$. Let $\mu\in\mathbb{N}$. Then the linear
equation

$\mu\sum_{i\in A}x_{i}=(k-\mu)\sum_{i\in B}x_{i}$

defines the $(A,B;\mu)$-hyperplane.

For any trivalent tree $T$ on $n$ leaves we consider an interior edge $e$. We
collect the labels on either sides of this edge $e$ in the sets $A$ and $B$.
These define a $(A,B,\mu=1)$ split-hyperplane.

The tree associated to a maximal cone gives a complete characterization of the
matroids appearing in the subdivision of the hypersimplex.

######  Theorem 7.

Let $T$ be a trivalent tree indexing a maximal cone of the Dressian. The cells
of the corresponding subdivision of the hypersimplex are indexed by the
internal vertices (non-leaves) of $T$. For each internal vertex $v\in V(T)$,
the corresponding cell contains the basis $ij$ if and only if the path from
$i$ to $j$ in $T$ passes through $v$.

###### Proof.

From [8] we know that a trivalent tree on $n$ leaves defines a compatible
system of splits for $\Delta(2,n)$. We show that the matroidal subdivision
that is obtained from the common refinements of this system of splits
coincides with the matroidal subdivision obtained from the indexing of the
trivalent tree $T$.

We consider the labelling $a_{1}\ldots a_{n}$ on $T$. Every internal edge of
$T$ provides a $(A,B)$ split of the form

$\displaystyle(A,B)_{e_{1}}=(\\{a_{1},a_{2}\\},\\{a_{3},\ldots\\})$
$\displaystyle\ldots\ldots$ $\displaystyle\ldots\ldots$
$\displaystyle(A,B)_{e_{n-3}}=(\\{a_{1},\ldots
a_{n-2}\\},\\{a_{n-1},a_{n}\\})$

Each split provides two matroid polytopes of the following form

$(S_{1})_{e_{1}}=\\{a_{1}a_{2},a_{1}a_{3},\ldots,a_{1}a_{n},a_{2}a_{3},\ldots
a_{2}a_{n}\\}$ $(S_{1})_{e_{2}}=\\{a_{1}a_{3},\ldots
a_{1}a_{n},a_{2}a_{3},\ldots a_{2}a_{n},\ldots a_{n}a_{n-1},\ldots
a_{3}a_{n}\\}$

The common refinements of all these n-3 splits provide us the following
matroidal subdivision $\Sigma$ with $n-2$ maximal cells,

$\displaystyle M_{1}=\\{a_{1}a_{2},\ldots,a_{2}a_{n}\\}\ldots$
$\displaystyle\ldots$ $\displaystyle
M_{n-2}=\\{a_{n-1}a_{1},\ldots,a_{n}a_{2}\\}$

As we see that these maximal cells are indexed by the $n-2$ internal vertices
of $T$. Hence, we see that $\Sigma$ coincides with the matroidal subdivision
obtained by the indexing of the tree $T$. ∎

We illustrate this with the following example for $\operatorname{\rm
Dr}(2,5)$.

######  Example 8 ($\operatorname{\rm Dr}(2,5)$).

We take the tree $T$ on 5 leaves as shown in Figure 2. The two internal edges
$e_{1}$ and $e_{2}$ define the following splits,

$(A,B)_{e_{1}}=\\{\\{1,2\\},\\{3,4,5\\}\\}$
$(A,B)_{e_{2}}=\\{\\{1,2,3\\},\\{4,5\\}\\}$

Since the points of the matroid polytopes reside on either side of the
hyperplanes defined by the corresponding split, we see that the matroid
polytopes defined by the split $(A,B)_{e_{1}}$ are

$S_{1}=\\{12,13,14,15,23,24,25\\}$ $S_{2}=\\{13,14,15,23,24,25,34,35,45\\}$

Similarly, for the split $(A,B)_{e_{2}}$

$S^{\prime}_{1}=\\{14,15,24,25,34,35,45\\}$
$S^{\prime}_{2}=\\{14,15,24,25,34,35,12,13,23\\}$

From [8] we know that such system of splits obtained from a tree are
compatible, and the common refinement of these two splits provides us the
following regular matroidal subdivision

$M_{1}=\\{12,13,14,15,23,24,25\\}$ $M_{2}=\\{13,14,15,23,24,25,34,35\\}$
$M_{3}=\\{14,15,24,25,34,35,45\\}$

which is the matroidal subdivision corresponding to the matroid with missing
Plücker coordinates {12,45}.

Figure 2: A trivalent tree for Dr(2,5)

######  Definition 9.

Let $n\geq 4$ and we consider a $n$-tuple of metric trees
$T=(T_{1},\ldots,T_{n})$, each with an associated metric $\delta_{i}$. We call
the $n$\- tuple $T$ of metric trees a _metric tree arrangement_ if,

$\delta_{i}(j,k)=\delta_{j}(i,k)=\delta_{k}(i,j)$

for all $i,j,k\in[n]$ pairwise distinct [7].

Moreover, considering trees $T_{i}$ without metrics, but with leaves still
labeled by $[n]\setminus{i}$, we say that $T$ is an _abstract tree
arrangement_ if,

  * •

$n=4$,

  * •

$n=5$, and $T$ is the set of quartets of a tree with five leaves,

  * •

$n\geq 6$, and $(T_{1}\setminus i,\ldots,T_{i-1}\setminus i,T_{i+1}\setminus
i,\ldots,T_{n}\setminus i)$ is an arrangement of $n-1$ trees for each
$i\in[n]$ [7].

Metric tree arrangements index cones of $Dr(3,n)$ and this is elaborated by
the following results in [7].

######  Lemma 10 (Lemma 4.2 in [7]).

Each matroid subdivision $\Sigma$ of $\Delta(3,n)$ defines an abstract
arrangement $T(\Sigma)$ of $n$ trees. Moreover, if $\Sigma$ is regular then
$T(\Sigma)$ supports a metric tree arrangement.

Two metric tree arrangements are _equivalent_ if they induce the same abstract
tree arrangement.

######  Theorem 11 (Theorem 4.4 [7] ).

The equivalence classes of arrangements of $n$ metric trees are in bijection
with the regular matroid subdivisions of the hypersimplex $\Delta(3,n)$.

##  3 Adjacency of cones in Dr(3,n)

###  3.1 Adjacency in cones in Dr(3,n)

We describe some notation for our results. Let $\Sigma^{\alpha}$ be a regular
matroidal subdivision indexing a cone $C$ in $\operatorname{\rm Dr}(3,n)$. We
refer $T^{\alpha}$ as the metric tree arrangement for the cone $C$. The
matroidal subdivision induced on the i-th contraction facet $\\{x_{i}=1\\}$ by
$\Sigma^{\alpha}$ is referred as $\Sigma^{\alpha}_{i}$.

######  Lemma 12.

Let $\Sigma^{\alpha}$ and $\Sigma^{\beta}$ be two regular matroidal
subdivisions of $\Delta(3,n)$ such that $\Sigma^{\alpha}$ and $\Sigma^{\beta}$
lie in adjacent cones of $\operatorname{\rm Dr}(3,n)$. Let $T^{\alpha}$ and
$T^{\beta}$ be the metric tree arrangements corresponding to $\Sigma^{\alpha}$
and $\Sigma^{\beta}$. Then, $\forall i\in[n]$ either,

  1. 1.

$\Sigma^{\alpha}_{i}=\Sigma^{\beta}_{i}$, or

  2. 2.

$\Sigma^{\alpha}_{i}$ and $\Sigma^{\beta}_{i}$ lie in adjacent cones of
$\operatorname{\rm Dr}(2,n-1)$

###### Proof.

By [7][Lemma 4.3] we know that each regular matroidal subdivision $\Sigma$ of
$\Delta(3,n)$ induces a regular matroidal subdivision $\Sigma_{i}$ on the i-th
contraction facet isomorphic to $\Delta(2,n-1)$. Let $T^{\alpha}_{i}$ be the
tree in $T^{\alpha}$ which is dual to the subdivision $\Sigma^{\alpha}_{i}$.

We proceed by contradiction and assume that there exists a set
$J\subsetneq[n]$, such that

$\Sigma^{\alpha}_{j}=\Sigma^{\beta}_{j},\quad\forall j\in J$

but for all $j^{\prime}\in[n]\setminus J$, $\Sigma^{\alpha}_{j^{\prime}}$ and
$\Sigma^{\beta}_{j^{\prime}}$ are not adjacent to each other, i.e., there
exists at least one cone $\Sigma^{\gamma}_{j"}$ (for some $\Sigma^{\gamma}\in
Dr(3,n)$) which lies in between $\Sigma^{\alpha}_{j^{\prime}}$ and
$\Sigma^{\beta}_{j^{\prime}}$ in $Dr(2,n-1)$. Since, adjacency in $Dr(2,n-1)$
is determined by a Whitehead move, hence $j"=j^{\prime}$, as all trees related
by a Whitehead move have the same set of leaves.

We now define a metric tree arrangement $T^{\gamma}$ such that

$T^{\alpha}_{j}=T^{\beta}_{j}=T^{\gamma}_{j},\quad\forall j\in J$

and for all $j^{\prime}\in[n]\setminus J$, the tree $T^{\alpha}_{j^{\prime}}$
is related by a Whitehead move to $T^{\gamma}_{j^{\prime}}$ and
$T^{\gamma}_{j^{\prime}}$ is related to $T^{\beta}_{j^{\prime}}$ by a
Whitehead move. By construction, $T^{\gamma}$ is not equivalent to the
abstract tree arrangements corresponding to $T^{\alpha}$ and $T^{\beta}$.
Hence, by [7][Theorem 4.5], $T^{\gamma}$ corresponds to the regular matroidal
subdivision $\Sigma^{\gamma}$. We now show that $\Sigma^{\gamma}$ is in
between $\Sigma^{\alpha}$ and $\Sigma^{\beta}$ to arrive at our desired
contradiction.

In [7], it is shown that for any metric tree arrangement $T$ in
$\operatorname{\rm Dr}(3,n)$, we can assign the following map

$\pi(i,j,k)=\begin{cases}\delta_{i}(j,k)=\delta_{j}(i,k)=\delta_{k}(i,j),&\text{if}\
\text{i,j and k pairwise distinct}\\\ \infty&\text{otherwise}\end{cases}$

to increasing tuples $i<j<k$ to obtain the finite tropical Pll̈ucker vector
which lies in the Plücker fan of $\operatorname{\rm Dr}(3,n)$. Hence, for the
tree arrangements $T^{\alpha},T^{\beta}$ and $T^{\gamma}$ we obtain the
vectors $\pi^{\alpha},\pi^{\beta}$ and $\pi^{\gamma}$. For
$\pi^{\alpha}\in\operatorname{\rm Dr}(3,n)$, this implies,

$\text{min}\\{\pi^{\alpha}_{hij}+\pi^{\alpha}_{hkl},\pi^{\alpha}_{hik}+\pi^{\alpha}_{hjl},\pi^{\alpha}_{hil}+\pi^{\alpha}_{hjk}\\}$

is attained at least twice for all pairwise distinct $h,i,j,k,l\in[n]$. But
this Plücker relation can be seen as the Plücker relation for the tree
$T_{h}\in T_{\alpha}$

$\text{min}\\{\pi^{\alpha}_{ij}+\pi^{\alpha}_{kl},\pi^{\alpha}_{ik}+\pi^{\alpha}_{jl},\pi^{\alpha}_{il}+\pi^{\alpha}_{jk}\\}$

which gives a point in $\operatorname{\rm Dr}(2,n-1)$. So for the $n$ trees in
$T^{\alpha}$ we have $n\choose 1$ choices of $h$, and for each such choice we
have $n\choose 4$ Plücker relations in $\operatorname{\rm Dr}(2,n-1)$. We know
that adjacency in $\operatorname{\rm Dr}(2,n-1)$ is determined by Whitehead
moves, so for all $j^{\prime}\in[n]\setminus J$, the initial forms
corresponding to Plücker relations for tree $T_{j^{\prime}}$ lie in adjacent
cones of $\operatorname{\rm Dr}(2,n-1)$ for $\pi^{\alpha}$ and $\pi^{\gamma}$.
But replacing $h$ in the indices we obtain the corresponding initial forms for
Plücker relations which determine points in adjacent cones in $Dr(3,n)$.
Hence, we conclude that $\pi^{\alpha}$ and $\pi^{\gamma}$ lie in adjacent
cones of the Plücker fan in $\operatorname{\rm Dr}(3,n)$. Similarly,
$\pi^{\beta}$ and $\pi^{\gamma}$ lie in adjacent cones of the Plücker fan in
$\operatorname{\rm Dr}(3,n)$. Therefore, the cone corresponding to
$\pi^{\gamma}$ is in between $\pi^{\alpha}$ and $\pi^{\beta}$. Since the
Plücker fan structure and secondary fan structure on the Dressian coincides
[11], this implies that the cone for $\Sigma^{\gamma}$ is in between
$\Sigma^{\alpha}$ and $\Sigma^{\beta}$, which gives us the contradiction.

∎

Hence, we now know the description of tree arrangements for adjacent cones in
$\operatorname{\rm Dr}(3,n)$. With this we now introduce the definition of
_generalized Whitehead moves_.

######  Definition 13.

Consider two metric tree arrangements $T^{\alpha}$ and $T^{\beta}$ containing
$m$ trees. Let $T^{\alpha}_{i}$ and $T^{\beta}_{i}$ denote the $i$-th metric
tree respectively. We say that $T^{\alpha}$ and $T^{\beta}$ are related via
generalized Whitehead move if, for a set $J\subset[m]$,

  * •

$T^{\alpha}_{j}=T^{\beta}_{j}\quad\forall j\in J$

  * •

$T^{\alpha}_{j^{\prime}}$ and $T^{\beta}_{j^{\prime}}$ is related by a
Whitehead move $\forall j^{\prime}\in[m]\setminus J$

With this definition we state the following result,

######  Theorem 14.

Two adjacent cones in $\operatorname{\rm Dr}(3,n)$ are related by generalized
Whitehead move.

###### Proof.

The prrof follows from Lemma 12 and Definition 13. ∎

This agrees with our computational observation for $\operatorname{\rm
Dr}(3,6)$ in Appendix A, where we see that adjacent cones have trees which are
either identical or related via a Whitehead move. This can be seen evidently
in the Figure 3, where we look at the tree arrangement on the wall between the
adjacent cones Cone9 and Cone10 in $\operatorname{\rm Dr}(3,6)$ which is
spanned by the ray generators $\\{42,53,54\\}$ and we see that the trees are
related by Whitehead moves, which shows up as trees with a degree four vertex.

Figure 3: Tree arrangement corresponding to the wall in between adjacent cones
Cone9 and Cone10 in $Dr(3,6)$

We now point the reader to some questions regarding adjacency of cones in
Dressian $\operatorname{\rm Dr}(3,n)$ that we would like to tackle in the
future,

  * •

Can the characterization of adjacency of cones in terms of Whitehead moves
between certain trees be extended to $Dr(k,n)$ for $k\geq 4$? In the next
section, with the introduction of generalized metric tree arrangements for
$\operatorname{\rm Dr}(k,n)$, a setup to try to prove such a result is
provided.

  * •

Can we bound the number of trees related via Whitehead move between two
adjacent cones for $\operatorname{\rm Dr}(3,n)$? In our example of
$\operatorname{\rm Dr}(3,6)$, this bound turns out to be three.

  * •

Can we obtain indices of the trees which are related via Whitehead moves
between adjacent cones, from only the information of the cones?

##  4 Generalized metric tree arrangements for Dr(k,n)

We begin this section with a discussion on the structure of Dressian
$\operatorname{\rm Dr}(k,n)$ ($k\geq 4)$. Firstly, we state Lemma 15 which is
a generalization of [7][Lemma 4.3],

######  Lemma 15.

Each matroid subdivision $\Sigma$ of $\Delta(k,n)$ defines an abstract tree
arrangement $T(\Sigma)$ of $\prescript{n\mkern-2.5mu}{}{P}_{k-2}$ trees, with
each tree having $n-(k-2)$ leaves. Moreover, if $\Sigma$ is regular then
$T(\Sigma)$ supports metric tree arrangement.

###### Proof.

Each of the $n$ contraction facets of $\Delta(k,n)$ are isomorphic to
$\Delta(k-1,n-1)$. Recursively using this relation we get that $\Sigma$
induces matroidal subdivisions on $(k-2)!\cdot{n\choose
k-2}=\prescript{n\mkern-2.5mu}{}{P}_{k-2}$ copies of $\Delta(2,n-(k-2))$. But
matroidal subdivisions of $\Delta(2,n-(k-2))$ are generated by compatible
system of splits and such subdivisions are dual to trees. Hence, $\Sigma$
gives rise to a tree arrangement.

Let $\Sigma$ be regular with $\lambda$ as a weight function. We now invoke
arguments, similar to what have been used in proof of Lemma 4.2 in [7]. By
suitable rescaling we can assume that $\lambda$ has values between $1$ and
$2$. We restrict $\lambda$ to $\Delta(2,n-(k-2))$ to obtain the induced
subdivisions. But a weight function on $\Delta(2,n-(k-2))$ which takes values
between $1$ and $2$ is a metric. Since, the induced subdivisions on
$\Delta(2,n-(k-2))$ are also regular matroidal subdivision, they are dual to
metric trees on $n-(k-2)$ leaves. The Split Decomposition theorem provides the
lengths on all edges of these trees. ∎

######  Example 16 (Metric tree arrangement for $\Delta(4,8)$).

We now describe the tree arrangement supported by a regular matroidal
subdivision of $\Delta(4,8)$. We obtain the following matroidal subdivision by
a lattice path construction,

$\begin{split}M_{1}&=\\{0,1,2,3,4,5,6,7,8,15,16,17,18,35,36,37,38\\}\\\
M_{2}&=\\{1,2,3,4,5,6,7,8,9,10,11,15,16,17,18,19,20,21,35,36,37,38,39,40,41\\}\\\
M_{3}&=\\{5,6,7,8,9,10,11,15,16,17,18,19,20,21,25,26,27,35,36,37,38,39,40,41,45,46,47\\}\\\
M_{4}&=\\{6,7,8,9,10,11,12,13,14,16,17,18,19,20,21,22,23,24,25,26,27,28,29\\\
&30,31,32,33,34,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50\\\
&51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69\\}\\\
M_{5}&=\\{15,16,17,18,25,26,27,35,36,37,38,45,46,47,55,56,57\\}\end{split}$

Using polymake [5], we find that this subdivision is induced by the following
weight vector

$\begin{split}w&=\\{0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,0,0,0,0,1,1,1,2,2,2,2,2,2,\\\
&3,3,3,4,4,4,5,0,0,0,0,1,1,1,2,2,2,2,2,2,3,3,3,\\\
&4,4,4,5,2,2,2,3,3,3,4,4,4,5,5,5,5,6,7\\}\end{split}$

The subdivision induced by this weight vector on the contraction facet
isomorphic to $\Delta(3,7)$ is given by the hyperplane of the form
$\\{x_{i}=1\\}$, and for $x_{1}=1$, the induced weight vector is

$\begin{split}w^{\text{in}}&=\\{0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,0,0,0,0,1,\\\
&1,1,2,2,2,2,2,2,3,3,3,4,4,4,5\\}\end{split}$

Figure 4: The tree arrangement for subdivision induced on a contraction facet
of $\Delta(4,8)$ together with the maximal cells of the corresponding matroid
subdivision.

With respect to a labeling of leaves with $[1,2\ldots 7]$ Figure 4 shows the
corresponding tree arrangement to this induced matroidal subdivision, along
with the corresponding maximal cells for the subdivision. The full metric tree
arrangement for the matroidal subdivision of $\Delta(4,8)$ is obtained by
considering all the other contraction facets, and obtaining the corresponding
tree arrangements on each copy of $\Delta(3,7)$. We also note that this
subdivision is coarser than one that corresponds to the maximal cones of
$Dr(4,8)$,, hence we see that not all trees in the tree arrangement are
trivalent.

We now describe an extension of the definition of metric tree arrangement [7].

######  Definition 17.

A generalised metric tree arrangement for $\operatorname{\rm Dr}(k,n)$ is a
set $T=\\{T_{I}\colon I\subseteq[n],\ |I|=k-2\\}$ of metric trees such that
the leaves of $T_{I}$ are labelled by the elements of $[n]\backslash I$ and
$T$ satisfies the following compatibility condition: for any pair of metric
trees $T_{I}$ and $T_{I^{\prime}}$ in $T$; any pair of leaves $i,j$ of
$T_{I}$; and any pair of leaves $i^{\prime},j^{\prime}$ of $T_{I^{\prime}}$,
if $I\cup\\{i,j\\}=I^{\prime}\cup\\{i^{\prime},j^{\prime}\\}$ then we have

$\delta_{I}(i,j)=\delta_{I^{\prime}}(i^{\prime},j^{\prime})$

where $\delta_{I}$ is the metric on $T_{I}$ and $\delta_{I^{\prime}}$ is the
metric on $T_{I^{\prime}}$. We define $\mathbb{T}(k,n)$ to be the collection
of all generalised metric tree arrangements for $\operatorname{\rm Dr}(k,n)$.

######  Remark 18.

It is evident from the definition that a metric tree arrangement is also a
generalised metric tree arrangement and corresponds to the case when $k=3$.

Each generalised metric tree $T$ induces a weight vector $w_{T}$ on the
Plücker variables given by $w_{T}(P_{K})=\delta_{J}(i,j)$ where:
$K=J\cup\\{i,j\\}$; $\delta_{J}$ is the metric on the tree $T_{J}$; and $i,j$
are leaves of the tree $T_{J}$. Note that the definition of the metric tree
arrangement above ensures that $w_{T}$ is a well-defined weight on the Plücker
variables. We establish the relation between the weight vector $w_{T}$ and the
Plücker fan structure of $\operatorname{\rm Dr}(k,n)$.

######  Theorem 19.

As sets, the collection of weights $\\{w_{T}\colon T\in\mathbb{T}(k,n)\\}$ is
equal to the Dressian $\operatorname{\rm Dr}(k,n)$.

###### Proof.

Let $T\in\mathbb{T}(k,n)$ be a tree arrangement. Fix a $3$-term tropical
Plücker relation for $\operatorname{\rm Dr}(k,n)$. Note that this relation is
of the form

$P=P_{J\cup ij}\odot P_{J\cup k\ell}\oplus P_{J\cup ik}\odot P_{J\cup
j\ell}\oplus P_{J\cup i\ell}\odot P_{J\cup jk}$

where $J\subseteq[n]$ has size $k-2$ and $i,j,k,\ell\in[n]\backslash J$ are
distinct. Let $S=\\{I\subseteq[n]\colon|I|=k,\ J\subseteq I\\}$ be the
$k$-subsets containing $J$. By definition, the restriction of $w_{T}$ to the
Plücker variables $P_{I}$ where $I\in S$ is induced by a single metric tree
$T_{J}\in T$. By [12], the space of weights induced by metric trees with
$n-k+2$ leaves equals the Dressian $\operatorname{\rm Dr}(2,n-k+2)$. Therefore
the maximum in $P$ is attained by at least two terms. So we have shown that
$\\{w_{T}\colon T\in\mathbb{T}(k,n)\\}\subseteq\operatorname{\rm Dr}(k,n)$.

For the other inclusion take any point $w\in\operatorname{\rm Dr}(k,n)$. For
each $(k-2)$-subset $J\subseteq[n]$, we construct a metric tree $T_{J}$ with
leaves $[n]\backslash J$. Consider the restriction of $w$ to the Plücker
variables $P_{I}$ where $J\subseteq I$. Note that these Plücker variables are
naturally in bijection with the $2$-subsets of $[n]\backslash J$. So, $w$
gives rise to point in $\operatorname{\rm Dr}(2,n-k+2)$ given by
$\hat{w}(P_{ij})=w(P_{J\cup ij})$. As above, the Dressian $\operatorname{\rm
Dr}(2,n-k+2)$ is equal to the space of weights induced by metric trees and so
we define $T_{J}$ to be the metric tree associated to $\hat{w}$ with leaves
labelled by $[n]\backslash J$. It remains to show that the trees satisfy the
compatibility condition. However, this is immediate since each metric tree
$T_{J}$ is constructed from the same point $w\in\operatorname{\rm Dr}(k,n)$. ∎

Alternatively, the result in Theorem 19 can be understood as a generalization
of the map $\pi$ defined in the proof of Theorem 4.4 in [7]. We describe this
perspective here. Let $\pi$ be the map from
$[n]^{k}\rightarrow\mathbb{R}\cup\\{\infty\\}$

$\pi(i_{1},i_{2},\ldots,i_{k})=\begin{cases}\delta_{I_{1}}(i_{1},i_{2})=\delta_{I_{1}}(i_{3},i_{4})\ldots=\delta_{I_{m}}(i_{k-1},i_{k})&i_{1},i_{2},\ldots,i_{k}\>\text{are
pairwise distinct}\\\ 0&\text{otherwise}\end{cases}$

where $I_{1},\ldots I_{m}$ are compatible sets of size $k-2$ satisfying
compatibility conditions described in Definition 17. With the proof of Theorem
19 we see that for a compatible set $I\in$ $[n]\choose k-2$, the minimum of

$\text{min}\\{\pi_{Ipq}+\pi_{Irs},\pi_{Ipr}+\pi_{Isq},\pi_{Ips}+\pi_{Iqr}\\}$

is attained twice, where $p,q,r,s\in[n]\setminus I$, as this minimum
corresponds to the weight induced by a metric tree on $n-k+2$ leaves. Hence,
we can conclude that the restriction of the map $\pi$ to increasing tuple
$i_{1}<i_{2}<\ldots<i_{k}$ is a finite tropical Plücker vector and is an
element of $Dr(k,n)$.

There is a natural fan structure on the space weights induced by generalised
metric tree arrangements. Two weights $w_{T}$ and $w_{T^{\prime}}$ lie in the
interior of the same cone if and only if $T$ and $T^{\prime}$ have the same
underlying set of labelled trees. The proof above naturally extends and shows
that the Plücker fan structure of the Dressian coincides with the fan
structure on the space of weights induced by generalised metric tree
arrangements. Moreover, the maximal cones of the Dressian are naturally
indexed by arrangements of trivalent trees. Also, as a consequence of Plücker
fan structure and secondary fan structure coinciding for the Dressian [11],
the fan structure on the space of weights induced by generalised metric tree
arrangements coincides with the secondary fan structure of Dressian.

The notion of a cherry of an arrangement naturally extends to the generalised
setting. A cherry of an arrangement $T\in\mathbb{T}(k,n)$ is a $k$-subset
$I\in[n]$ such that if $I=J\cup\\{i,j\\}$ is any partition then $i,j$ is a
cherry in tree $T_{J}$.

With Theorem 19 we see that a generalized metric tree arrangement corresponds
to a regular matroidal subdivision of $\Delta(k,n)$. We see this when we
consider the collection of weights $\\{w_{T}\colon T\in\mathbb{T}(k,n)\\}$ as
a height function on the Plücker coordinates. Using the fact that the Plücker
fan structure and secondary fan structure coincide for the Dressian, we obtain
the following result,

######  Corollary 20.

Each regular matroidal subdivision $\Sigma$ of $\Delta(k,n)$ supports a
generalized metric tree arrangement.

###### Proof.

Let us consider the weight function for the regular matroidal subdivision
$\Sigma$ to be $w_{\Sigma}$ and $w_{\Sigma}\in\operatorname{\rm Dr}(k,n)$.
Since the Plücker fan structure and secondary fan structure coincide for the
Dressian, therefore by Theorem 19, we can conclude that there exists a
generalized metric tree arrangement $T(k,n)$ such that it induces
$w_{\Sigma}$. ∎

We know that in [7] Lemma 4.2 and Theorem 4.5 help establish a bijection
between equivalence classes of _equivalent_ metric tree arrangements of $n$
trees and regular matroidal subdivisions of $\Delta(3,n)$. We now try to
establish this equivalence for generalized metric tree arrangements. We
already see a mapping from one side, i.e., by Theorem 19 we see that each
generalized metric tree arrangement corresponds to a regular matroidal
subdivision of $\Delta(k,n)$. We now have to show a mapping in the reverse
direction.

We refer to a generalized metric tree arrangement as an _abstract_ generalized
metric tree arrangements if we consider the tree arrangement without the
metric. Two generalized metric tree arrangements are _equivalent_ if they
induce the same abstract tree arrangement. With these results we can now state
the following result which in essence is a generalisation of Theorem 4.4 in
[7].

######  Corollary 21.

The equivalence classes of arrangements of $n\choose k-2$ generalized metric
tree are in bijection with the regular matroid subdivisions of $\Delta(k,n)$.

###### Proof.

With Theorem 19, we see that a generalized metric tree arrangement defines a
point in the Dressian $\operatorname{\rm Dr}(k,n)$. We observe that equivalent
generalized metric tree arrangements define points in the $\operatorname{\rm
Dr}(k,n)$ which lie in the same cone in the Plücker fan. Considering this
weight vector as a lifting function provides us the corresponding regular
matroid subdivision. This provides us a map from equivalence classes of
generalized metric tree arrangements to regular matroidal subdivisions. For
the map from other side we consider a regular matroidal subdivisions of
$\operatorname{\rm Dr}(k,n)$ and by Corollary 20 we see that this subdivision
corresponds to a generalized metric tree arrangement. By [11], we know that
the secondary fan structure and the Plücker fan structure coincides for
$\operatorname{\rm Dr}(k,n)$. Hence, this gives us the required bijection. ∎

It is important to notice that for $Dr(k,n),k\geq 4$ we encounter both metric
tree arrangements and generalized metric tree arrangements. However, we see
that the metric arrangement obtained in Lemma 15 consists of $(k-2)!$ $\cdot
n\choose k-2$ whereas the generalized metric tree arrangement consists of
$n\choose k-2$ trees.

######  Remark 22.

For $k=3$, we see that $(k-2)!$ $\cdot n\choose k-2$ = $n\choose k-2$,
therefore the associated metric tree arrangement and generalized metric tree
arrangement coincides in this specific case.

Our study leads us to the following questions which we aim to explore in
subsequent work,

  * •

Compute the Dressian $\operatorname{\rm Dr}(4,8)$, which could provide
computational examples to complement our results. Although at the moment the
state-of-the-art in terms of such computations is $\operatorname{\rm Dr}(3,8)$
[9],[2].

  * •

We are able to provide the notion of metric tree arrangements for all values
of $\operatorname{\rm Dr}(k,n)$. This leads us to ask that whether we can find
an equivalent definition of a tree arrangement for the case of local Dressian
$\operatorname{\rm Dr}(M)$, where $M$ is a matroid. In recent work [3] there
have been advancements in techniques to compute $Dr(M)$ for certain matroids
which could be beneficial for answering this question.

We condense some of the findings in this section with this statement,

######  Corollary 23.

Each maximal cone of the Dressian $Dr(k,n),n\geq 4,k\geq 2$ supports an
arrangement of $n\choose k-2$ metric trees.

#### Acknowledgment

I am grateful to my advisor Prof Isabel Van Driessche and also to Prof Tom De
Medts for their support during the preparation of this article. I would also
like to thank Prof Michael Joswig for going through earlier drafts of this
work. I thank Dr. Ahmed Umer Ashraf for discussions which helped in coming up
with the examples in the text. A. K. Tewari has been supported by the UGent
BOF grant BOF/STA/201909/038 and Ugent BOF grant BOFFF12015001101.

## References

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##  Appendix A Appendix

###  A.1 Metric tree arrangements for Dr(3,6)

For Dressian $\operatorname{\rm Dr}(3,6)$ there are, up to symmetry, $7$
different maximal cones. We now list the seven cones along with the
descriptions of the metric tree arrangements and the associated regular
matroidal subdivsion.

For cone number $0$ from Jensen’s website, the tree arrangement corresponding
to this cone is

$\displaystyle T_{1}:C(25,4,36),\quad T_{2}:C(15,3,46),\quad
T_{3}:C(16,2,45),$ $\displaystyle T_{4}:C(26,1,35),\quad
T_{5}:C(12,6,34),\quad T_{6}:C(13,5,24)$

where $C(ab,c,de)$ is the caterpillar graph with cherries $ab$ and $cd$. For
each tree $T_{i}=(a_{i}b_{i},c_{i},d_{i}e_{i})$ we write: $s_{i}$ for the
internal vertex of $T_{i}$ adjacent to $a_{i}$ and $b_{i}$; $t_{i}$ for the
internal vertex of $T_{i}$ adjacent to $c_{i}$; and $u_{i}$ for the internal
vertex of $T_{i}$ adjacent to $d_{i}$ and $e_{i}$.

The matroid subdivision induced by this cone is given by $5$ cells whose
corresponding matroids are $M_{A},M_{B},M_{C},M_{D}$ and $M_{E}$: see Figure
5.

Figure 5: The tree arrangement for cone $0$ together with the maximal cells of
the corresponding matroid subdivision.

######  Definition 24.

Given a compatible tree arrangement $T=(T_{1},\dots T_{n})$ a cherry of $T$ is
a subset $\\{i,j,k\\}\in\binom{[n]}{3}$ such that: $ij$ is a cherry in
$T_{k}$; $ik$ is a cherry in $T_{j}$; and $jk$ is a cherry in $T_{i}$

Cone Index |  List of Cherries
---|---
$C_{0}$ |  $\\{125\\},\\{136\\},\\{246\\},\\{345\\}$
$C_{1}$ |  $\\{246\\},\\{345\\},\\{125\\}$
$C_{2}$ |  $\\{125\\},\\{246\\}$
$C_{3}$ |  $\\{125\\},\\{246\\}$
$C_{4}$ |  $\\{135\\},\\{246\\}$
$C_{5}$ |  $\\{246\\}$
$C_{6}$ |  $\\{\\}$
Table 1: List of cones from Jensen’s website and their associated cherries

For Cone $6$ from Jensen’s website, the tree arrangement is as follows,

$\displaystyle T_{1}:C(23,5,46),\quad T_{2}:C(15,3,46),\quad
T_{3}:C(15,2,46),$ $\displaystyle T_{4}:C(15,6,23),\quad
T_{5}:C(23,1,46),\quad T_{6}:C(15,4,23)$

Figure 6: The tree arrangement for cone $6$ together with the maximal cells of
the corresponding matroid subdivision.

and the tree arrangement along with the matroidal subdivision is shown in the
Figure 6.

For Cone $1$ from Jensen’s website, the tree arrangement is as follows,

$\displaystyle T_{1}:C(25,6,34),\quad T_{2}:C(15,3,46),\quad
T_{3}:C(26,1,45),$ $\displaystyle T_{4}:C(26,1,35),\quad
T_{5}:C(12,6,34),\quad T_{6}:C(15,3,24)$

Figure 7: The tree arrangement for cone $1$ together with the maximal cells of
the corresponding matroid subdivision.

and the tree arrangement along with the matroidal subdivision is shown in the
Figure 7.

For Cone $2$ from Jensen’s website, the tree arrangement is as follows,

$\displaystyle T_{1}:C(25,6,34),\quad T_{2}:C(15,3,46),\quad
T_{3}:C(15,4,26),$ $\displaystyle T_{4}:C(15,3,26),\quad
T_{5}:C(12,6,34),\quad T_{6}:C(15,3,24)$

Figure 8: The tree arrangement for cone $2$ together with the maximal cells of
the corresponding matroid subdivision.

and the tree arrangement along with the matroidal subdivision is shown in the
Figure 8.

For Cone $3$ from Jensen’s website, the tree arrangement is as follows,

$\displaystyle T_{1}:C(25,3,46),\quad T_{2}:C(15,3,46),\quad
T_{3}:C(15,2,46),$ $\displaystyle T_{4}:C(15,3,26),\quad
T_{5}:C(12,3,46),\quad T_{6}:C(15,3,24)$

Figure 9: The tree arrangement for cone $3$ together with the maximal cells of
the corresponding matroid subdivision.

and the tree arrangement along with the matroidal subdivision is shown in the
Figure 9.

For Cone $4$ from Jensen’s website, the tree arrangement is as follows,

$\displaystyle T_{1}:C(35,2,46),\quad T_{2}:C(15,3,46),\quad
T_{3}:C(15,2,46),$ $\displaystyle T_{4}:C(15,3,26),\quad
T_{5}:C(13,2,46),\quad T_{6}:C(15,3,24)$

Figure 10: The tree arrangement for cone $4$ together with the maximal cells
of the corresponding matroid subdivision.

and the tree arrangement along with the matroidal subdivision is shown in the
Figure 10.

For Cone $5$ from Jensen’s website, the tree arrangement is as follows,

$\displaystyle T_{1}:C(23,5,46),\quad T_{2}:C(15,3,46),\quad
T_{3}:C(15,2,46),$ $\displaystyle T_{4}:C(15,3,26),\quad
T_{5}:C(23,1,46),\quad T_{6}:C(15,3,24)$

Figure 11: The tree arrangement for cone $5$ together with the maximal cells
of the corresponding matroid subdivision.

and the tree arrangement along with the matroidal subdivision is shown in the
Figure 11.

###  A.2 Adjacent cone tree arrangements in Dr(3,6)

In this section we try to look at what happens if we consider tree
arrangements of two adjacent cones in $\operatorname{\rm Dr}(3,6)$. For this
we do a computational study using Macaulay2 [6] and consider the
$Cone_{0}=\\{42,52,53,54\\}$ (missing Plücker coordinate = $\\{123\\}$) where
the indices represent the rays indexing an interior point in the cone. We
compute all the cones adjacent to $Cone_{0}$ in $\operatorname{\rm Dr}(3,6)$
and compute their respective tree arrangements. $Cone_{0}$ lies in the
equivalence class of $Cone_{5}$ in the list on Jensen’s website. There are
eight adjacent cones to $Cone_{0}$ and Table 2 lists them with the rays
indexing each of them and also the missing Pll̈ucker coordinates. The missing
Pll̈ucker coordinates for each cone are exactly the _cherries_ of the tree
arrangement.

Cone Index |  List of Rays |  Cherries / Missing Plücker Coordinates
---|---|---
$C_{9}$ |  $\\{15,42,53,54\\}$ |  $\\{123,256\\}$
$C_{10}$ |  $\\{12,42,53,54\\}$ |  $\\{123,156\\}$
$C_{11}$ |  $\\{14,42,52,54\\}$ |  $\\{123,346\\}$
$C_{12}$ |  $\\{13,42,52,54\\}$ |  $\\{123,345\\}$
$C_{81}$ |  $\\{52,53,54,64\\}$ |  $\\{123,456\\}$
$C_{90}$ |  $\\{20,52,53,54\\}$ |  $\\{123,356\\}$
$C_{105}$ |  $\\{25,42,52,53,63\\}$ |  $\\{\\}$
$C_{114}$ |  $\\{42,43,52,53\\}$ |  $\\{124\\}$
Table 2: List of adjacent cones for $Cone_{0}$

The tree arrangement for $Cone_{0}$ is shown in Figure 12. It is obtained by
first computing the contraction of all maximal cells of the matroidal
subdivision $\Sigma$ for the cone with respect to each index $\\{i\\}$ in
$[n]$. From the structure of matroidal subdivision on $\Delta(2,5)$, we know
that the cell indexed by an internal vertex attached to a cherry of a tree
$T_{i}$ has 7 elements, while the middle internal vertex has 8 elements.
Hence, for each contraction on $\Sigma$, we obtain three unique cells
$M_{a}/\\{i\\},M_{b}/\\{i\\}$ and $M_{c}/\\{i\\}$, out of which two have 7
elements and one has 8 elements. If the 7 element cell indexes a vertex
attached to a cherry $pq$ then all elements in the cell contain either $p$ or
$q$. With this we obtain labels for the cherry of each tree $T_{i}$. The label
for the middle leaf is the one index left from $[6]$ after labeling the
cherries of the tree and the tree index.

Figure 12: The tree arrangement for $Cone_{0}$ together with the maximal cells
of the corresponding matroid subdivision.

For the adjacent $Cone_{9}$, we have the tree arrangement shown in Figure 13

Figure 13: The tree arrangement for $Cone_{9}$ together with the maximal cells
of the corresponding matroid subdivision. The shaded edges represent the
labels which get changed with respect to the labeling in $Cone_{0}$

For the adjacent Cone10, we have the tree arrangement shown in Figure 14

Figure 14: The tree arrangement for Cone10 together with the maximal cells of
the corresponding matroid subdivision. The shaded edges represent the labels
which get changed with respect to the labeling in $Cone_{0}$

For the adjacent Cone11, we have the tree arrangement shown in Figure 15

Figure 15: The tree arrangement for Cone11 together with the maximal cells of
the corresponding matroid subdivision. The shaded edges represent the labels
which get changed with respect to the labeling in $Cone_{0}$

For the adjacent Cone12, we have the tree arrangement shown in Figure 16

Figure 16: The tree arrangement for Cone12 together with the maximal cells of
the corresponding matroid subdivision. The shaded edges represent the labels
which get changed with respect to the labeling in $Cone_{0}$

For the adjacent Cone81, we have the tree arrangement shown in Figure 17

Figure 17: The tree arrangement for Cone81 together with the maximal cells of
the corresponding matroid subdivision. The shaded edges represent the labels
which get changed with respect to the labeling in $Cone_{0}$

For the adjacent Cone90, we have the tree arrangement shown in Figure 18

Figure 18: The tree arrangement for Cone90 together with the maximal cells of
the corresponding matroid subdivision. The shaded edges represent the labels
which get changed with respect to the labeling in $Cone_{0}$

For the adjacent Cone105, we have the tree arrangement shown in Figure 19

Figure 19: The tree arrangement for Cone105 together with the maximal cells of
the corresponding matroid subdivision. The shaded edges represent the labels
which get changed with respect to the labeling in $Cone_{0}$

For the adjacent Cone114, we have the tree arrangement shown in Figure 20

Figure 20: The tree arrangement for Cone114 together with the maximal cells of
the corresponding matroid subdivision. The shaded edges represent the labels
which get changed with respect to the labeling in $Cone_{0}$

Hence, we see that for $Cone_{0}$, adjacent cones have tree arrangements
wherein all trees are identical except two trees which are related by a
Whitehead move. We conduct this computational study for cones of every
equivalence class out of the seven distinct classes of cones for $Dr(3,6)$. We
observe that in all of them, for adjacent cones, all trees are identical
except at most three tree which are related via a Whitehead move. The
computational files for the other six equivalence classes can be found at

https://github.com/Ayush-Tewari13/Cone_Adjacency_Dr-3-6-