Search NASA⌕ Search

SEARCH · Search NASA

Results for “tree”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 775 records · Page 43

A Multi-Encoding Approach for LTL Symbolic Satisfiability Checking

Formal behavioral specifications written early in the system-design process and communicated across all design phases have been shown to increase the efficiency, consistency, and quality of the system under development. To prevent introducing design or verification errors, it is crucial to test specifications for satisfiability. Our focus here is on specifications expressed in linear temporal logic (LTL). We introduce a novel encoding of symbolic transition-based Buchi automata and a novel, "sloppy," transition encoding, both of which result in improved scalability. We also define novel BDD variable orders based on tree decomposition of formula parse trees. We describe and extensively test a new multi-encoding approach utilizing these novel encoding techniques to create 30 encoding variations. We show that our novel encodings translate to significant, sometimes exponential, improvement over the current standard encoding for symbolic LTL satisfiability checking.

Rozier, Kristin Y.↗

Oscillation of Angiogenesis with Vascular Dropout in Diabetic Retinopathy by VESsel GENeration Analysis (VESGEN)

PURPOSE. Vascular dropout and angiogenesis are hallmarks of the progression of diabetic retinopathy (DR). However, current evaluation of DR relies on grading of secondary vascular effects, such as microaneurysms and hemorrhages, by clinical examination instead of by evaluation of actual vascular changes. The purpose of this study was to map and quantify vascular changes during progression of DR by VESsel GENeration Analysis (VESGEN). METHODS. In this prospective cross-sectional study, 15 eyes with DR were evaluated with fluorescein angiography (FA) and color fundus photography, and were graded using modified Early Treatment Diabetic Retinopathy Study criteria. FA images were separated by semiautomatic image processing into arterial and venous trees. Vessel length density (L(sub v)), number density (N(sub v)), and diameter (D(sub v)) were analyzed in a masked fashion with VESGEN software. Each vascular tree was automatically segmented into branching generations (G(sub 1)...G(sub 8) or G(sub 9)) by vessel diameter and branching. Vascular remodeling status (VRS) for N(sub v) and L(sub v) was graded 1 to 4 for increasing severity of vascular change. RESULTS. By N(sub v) and L(sub v), VRS correlated significantly with the independent clinical diagnosis of mild to proliferative DR (13/15 eyes). N(sub v) and L(sub v) of smaller vessels (G(sub >=6) increased from VRS1 to VRS2 by 2.4 X and 1.6 X, decreased from VRS2 to VRS3 by 0.4 X and 0.6X, and increased from VRS3 to VRS4 by 1.7 X and 1.5 X (P < 0.01). Throughout DR progression, the density of larger vessels (G(sub 1-5)) remained essentially unchanged, and D(sub v1-5) increased slightly. CONCLUSIONS. Vessel density oscillated with the progression of DR. Alternating phases of angiogenesis/neovascularization and vascular dropout were dominated first by remodeling of arteries and subsequently by veins.

Parsons-Wingerter, Patricia↗

A First-Order Radiative Transfer Model for Microwave Radiometry of Forest Canopies at L-Band

In this study, a first-order radiative transfer (RT) model is developed to more accurately account for vegetation canopy scattering by modifying the basic Tau-Omega model (the zero-order RT solution). In order to optimally utilize microwave radiometric data in soil moisture (SM) retrievals over vegetated landscapes, a quantitative understanding of the relationship between scattering mechanisms within vegetation canopies and the microwave brightness temperature is desirable. The first-order RT model is used to investigate this relationship and to perform a physical analysis of the scattered and emitted radiation from vegetated terrain. This model is based on an iterative solution (successive orders of scattering) of the RT equations up to the first order. This formulation adds a new scattering term to the . model. The additional term represents emission by particles (vegetation components) in the vegetation layer and emission by the ground that is scattered once by particles in the layer. The model is tested against 1.4-GHz brightness temperature measurements acquired over deciduous trees by a truck-mounted microwave instrument system called ComRAD in 2007. The model predictions are in good agreement with the data, and they give quantitative understanding for the influence of first-order scattering within the canopy on the brightness temperature. The model results show that the scattering term is significant for trees and modifications are necessary to the . model when applied to dense vegetation. Numerical simulations also indicate that the scattering term has a negligible dependence on SM and is mainly a function of the incidence angle and polarization of the microwave observation. Index Terms Emission,microwave radiometry, scattering, soil, vegetation.

Kurum, Mehmet↗

The Potential Impact of CO2 and Air Temperature Increases on Krummholz's Transformation into Arborescent Form in the Southern Siberian Mountains

Trees in the southern Siberian Mountains forest-tundra ecotone have considerably increased their radial and apical growth increments during the last few decades. This leads to the widespread vertical transformation of mat and prostrate krummholz forms of larch (Larix sibirica Ledeb) and Siberian pine (Pinus sibirica Du Tour). An analysis of the radial growth increments showed that these transformations began in the mid-1980s. Larch showed a greater resistance to the harsh alpine environment and attained a vertical growth form in areas where Siberian pine is still krummholz. Upper larch treeline is about 10 m higher than Siberian pine treeline. Observed apical and radial growth increment increases were correlated with CO2 concentration (r = 0.83-0.87), summer temperatures (r = 0.55-0.64), and "cold period" (i.e. September-May) air temperatures (r = 0.36-0.37). Positive correlation between growth increments and winter precipitation was attributed to snow cover protection for trees during wintertime.

Kharuk, V. I.↗

Wide Range SET Pulse Measurement

A method for measuring a wide range of SET pulses is demonstrated. Use of dynamic logic, faster than ordinary CMOS, allows capture of short pulses. A weighted binning of SET lengths allows measurement of a wide range of pulse lengths with compact circuitry. A pulse-length-conservative pulse combiner tree routes SETs from combinational logic to the measurement circuit, allowing SET measurements in circuits that cannot easily be arranged in long chains. The method is applied to add-multiplex combinational logic, and to an array of NFET routing switches, at .35 micron. Pulses are captured in a chain of Domino Logic AND gates. Propagation through the chain is frozen on the trailing edge by dropping low the second "enable" input to the AND gates. Capacitive loading is increased in the latter stages to create an approximately logarithmic weighted binning, so that a broad range of pulse lengths can be captured with a 10 stage capture chain. Simulations show pulses can be captured which are 1/5th the length of those typically captured with leading edge triggered latch methods, and less than the length of those captured with a trailing edge latch method. After capture, the pulse pattern is transferred to an SEU protected shift register for readout. 64 instances of each of two types of logic are used as targets. One is a full adder with a 4 to 1 mux on its inputs. The other is a 4 x 4 NFET routing matrix. The outputs are passed through buffered XNOR comparators to identify pulses, which are merged in a buffered not-nand (OR) tree designed to avoid pulse absorption as much as possible. The output from each of the two test circuits are input into separate pulse measurement circuits. Test inputs were provided so that the circuit could be bench tested and calibrated. A third SET measurement circuit with no inputs was used to judge the contribution from direct hits on the measurement circuit. Heavy ions were used with an LET range from 12 to 176. At LET of 21 and below, the very small number of SETs were not significantly higher in the test over the control circuits. At higher LET the test circuit SETs are one or two orders of magnitude greater than for the control circuit. The NFET circuit produces more and slightly longer SETs as expected. But the differences do not appear to be significant enough to modify strategies now used to avoid capture of SETs in chips such as FPGAs. Complete data and graphs will be in the full paper / presentation. In the summary figure below left, NOCL is the reference circuit without any input, and number of stages triggered is plotted. Simulation at right shows the smallest pulse captured (stage 2) at about 300 ps. Our conclusion is that the method is promising, but that improvements in the merge network are desirable before applying in a deep submicron process

Shuler, Robert L.↗

Wildfires in Northern Siberian Larch Dominated Communities

The fire history of the northern larch forests within the permafrost zone in a portion of northern Siberia (approx 66 deg N, 100 deg E) was studied. Since there is little to no human activities in this area fires within the study area were mostly caused by lightning. Fire return intervals (FRI) were estimated based on burn marks on tree stems and dates of tree natality. FRI values varied from 130 yr to 350 yr with 200 +/- 50 yr mean. In southerly larch dominated communities FRI was found to be shorter (77 +/- 20 yr at approx 61 deg. N, and 82 +/- 7 at 64 deg N), and longer at the northern boundary (approx 71 deg) of larch stands (320 +/- 50 yr). During the Little Ice Age period in the 16th to 18th centuries FRI was approximately twice as long as recorded in this study. Fire caused changes in the soil including increases in soil drainage and permafrost thawing depth and a radial growth increase of about 2 times (with more than 6 times observed). This effect may simulate the predicted warming impact on the larch growth in the permafrost zone.

Khaurk, Viacheslav I.↗

Space Flight Middleware: Remote AMS over DTN for Delay-Tolerant Messaging

This paper describes a technique for implementing scalable, reliable, multi-source multipoint data distribution in space flight communications -- Delay-Tolerant Reliable Multicast (DTRM) -- that is fully supported by the "Remote AMS" (RAMS) protocol of the Asynchronous Message Service (AMS) proposed for standardization within the Consultative Committee for Space Data Systems (CCSDS). The DTRM architecture enables applications to easily "publish" messages that will be reliably and efficiently delivered to an arbitrary number of "subscribing" applications residing anywhere in the space network, whether in the same subnet or in a subnet on a remote planet or vehicle separated by many light minutes of interplanetary space. The architecture comprises multiple levels of protocol, each included for a specific purpose and allocated specific responsibilities: "application AMS" traffic performs end-system data introduction and delivery subject to access control; underlying "remote AMS" directs this application traffic to populations of recipients at remote locations in a multicast distribution tree, enabling the architecture to scale up to large networks; further underlying Delay-Tolerant Networking (DTN) Bundle Protocol (BP) advances RAMS protocol data units through the distribution tree using delay-tolerant storeand- forward methods; and further underlying reliable "convergence-layer" protocols ensure successful data transfer over each segment of the end-to-end route. The result is scalable, reliable, delay-tolerant multi-source multicast that is largely self-configuring.

Disruption Tolerant Networking (DTN)↗

Modeling the Height of Young Forests Regenerating from Recent Disturbances in Mississippi using Landsat and ICESat data

Many forestry and earth science applications require spatially detailed forest height data sets. Among the various remote sensing technologies, lidar offers the most potential for obtaining reliable height measurement. However, existing and planned spaceborne lidar systems do not have the capability to produce spatially contiguous, fine resolution forest height maps over large areas. This paper describes a Landsat-lidar fusion approach for modeling the height of young forests by integrating historical Landsat observations with lidar data acquired by the Geoscience Laser Altimeter System (GLAS) instrument onboard the Ice, Cloud, and land Elevation (ICESat) satellite. In this approach, "young" forests refer to forests reestablished following recent disturbances mapped using Landsat time-series stacks (LTSS) and a vegetation change tracker (VCT) algorithm. The GLAS lidar data is used to retrieve forest height at sample locations represented by the footprints of the lidar data. These samples are used to establish relationships between lidar-based forest height measurements and LTSS-VCT disturbance products. The height of "young" forest is then mapped based on the derived relationships and the LTSS-VCT disturbance products. This approach was developed and tested over the state of Mississippi. Of the various models evaluated, a regression tree model predicting forest height from age since disturbance and three cumulative indices produced by the LTSS-VCT method yielded the lowest cross validation error. The R(exp 2) and root mean square difference (RMSD) between predicted and GLAS-based height measurements were 0.91 and 1.97 m, respectively. Predictions of this model had much higher errors than indicated by cross validation analysis when evaluated using field plot data collected through the Forest Inventory and Analysis Program of USDA Forest Service. Much of these errors were due to a lack of separation between stand clearing and non-stand clearing disturbances in current LTSS-VCT products and difficulty in deriving reliable forest height measurements using GLAS samples when terrain relief was present within their footprints. In addition, a systematic underestimation of about 5 m by the developed model was also observed, half of which could be explained by forest growth that occurred between field measurement year and model target year. The remaining difference suggests that tree height measurements derived using waveform lidar data could be significantly underestimated, especially for young pine forests. Options for improving the height modeling approach developed in this study were discussed.

Li, Ainong↗

How Did I Get Here?

What country was responsible for much of the early work in laser altimetry, who was the first scientist to look at a vegetation profile generated by a laser, who first related laser ranges to forest structure? The purpose of this paper is to look back 3+ decades ago when forestry lidar was in its formative years and to give credit to those who first thought to employ a laser to measure the Earth's surface. As with much scientific research, advances were often made independently and concurrently in a number of countries. Functioning lasers were first demonstrated in 1960 in the USA and in 1961 in the USSR, but research into the use of lasers as forest measurement tools did not begin for another IS years. Initially, with respect to Earth resources, lasers were employed to measure sea ice surface roughness, to make near-shore bathymetric measurements, to penetrate forests to make detailed topographic measurements, and to fluoresce oceanic phytoplankton for surface current studies. Some of these early studies noted that forest profiles were evident but in fact added noise to topographic retrievals. Trees became the signal rather than noise in 1976 when researchers in the Canadian Forest Service went to an IUFRO conference in Oslo and reported on efforts to better estimate timber volume in tropical forests. They employed large-scale airphotos (1:500 up to 1:4000) to measure the top of the tropical canopy, an avionic radar to measure the location of the ground beneath the canopy, and a barometric sensor to record aircraft/radar height above a datum (e.g., sea level). Their radar did not work well in dense vegetation and mentioned in passing that a laser altimeter had worked well in Canadian forests. But in a report 2 years later they reported that, on a study area in Costa Rica, the radar ground line was much improved and no mention is made of a lidar altimeter. In the USSR in 1977, Russian researchers felled a birch and a spruce, aimed a helium-neon, 0.63 micron laser with a spot size of approx 25mm, at the horizontal trees, produced a profilograph, compared it to tape measurements, and concluded that, with increased power, such a laser could be mounted on an aircraft to remotely measure forest canopies. In 1979, they mounted their He-Ne laser on an AN-2 biplane and acquired their fIrSt airborne profiles. Scientists with TRANARG, a mapping/surveying company in Caracas, Venezuela, reported on 1976 flights employing a helium-neon lidar to collect over 11,000 km of lidar profiles spaced 1.5 km apart in order to construct a topographic map to help site a new reservoir. They noted 35-40 m median canopy heights with emergents up to 55 m in their profiles. These studies and others that utilize these height and canopy density metrics for forest mensuration are reviewed in this glance backwards at the history of forestry lidar.

Nelson, Ross↗

An Automated Algorithm to Screen Massive Training Samples for a Global Impervious Surface Classification

An algorithm is developed to automatically screen the outliers from massive training samples for Global Land Survey - Imperviousness Mapping Project (GLS-IMP). GLS-IMP is to produce a global 30 m spatial resolution impervious cover data set for years 2000 and 2010 based on the Landsat Global Land Survey (GLS) data set. This unprecedented high resolution impervious cover data set is not only significant to the urbanization studies but also desired by the global carbon, hydrology, and energy balance researches. A supervised classification method, regression tree, is applied in this project. A set of accurate training samples is the key to the supervised classifications. Here we developed the global scale training samples from 1 m or so resolution fine resolution satellite data (Quickbird and Worldview2), and then aggregate the fine resolution impervious cover map to 30 m resolution. In order to improve the classification accuracy, the training samples should be screened before used to train the regression tree. It is impossible to manually screen 30 m resolution training samples collected globally. For example, in Europe only, there are 174 training sites. The size of the sites ranges from 4.5 km by 4.5 km to 8.1 km by 3.6 km. The amount training samples are over six millions. Therefore, we develop this automated statistic based algorithm to screen the training samples in two levels: site and scene level. At the site level, all the training samples are divided to 10 groups according to the percentage of the impervious surface within a sample pixel. The samples following in each 10% forms one group. For each group, both univariate and multivariate outliers are detected and removed. Then the screen process escalates to the scene level. A similar screen process but with a looser threshold is applied on the scene level considering the possible variance due to the site difference. We do not perform the screen process across the scenes because the scenes might vary due to the phenology, solar-view geometry, and atmospheric condition etc. factors but not actual landcover difference. Finally, we will compare the classification results from screened and unscreened training samples to assess the improvement achieved by cleaning up the training samples. Keywords:

Tan, Bin↗

Use of Windbreaks for Hurricane Protection of Critical Facilities

The protection of NASA Langley Research Center from future hurricanes is important in order to allow the center to fulfill its mission. The impact of the center is not only great within NASA but the economy as well. The infrastructure of the Center is under potential risk in the future because of more intense hurricanes with higher speed winds and flooding. A potential method of protecting the Center s facilities is the placement of a windbreak barrier composed of indigenous trees. The New Town program that is now in progress creates a more condensed area of focus for protection. A potential design for an efficient tree windbreak barrier for Langley Research center is proposed.

Hyater-Adams, Sinone↗

Evaluation of Algorithms for a Miles-in-Trail Decision Support Tool

Four machine learning algorithms were prototyped and evaluated for use in a proposed decision support tool that would assist air traffic managers as they set Miles-in-Trail restrictions. The tool would display probabilities that each possible Miles-in-Trail value should be used in a given situation. The algorithms were evaluated with an expected Miles-in-Trail cost that assumes traffic managers set restrictions based on the tool-suggested probabilities. Basic Support Vector Machine, random forest, and decision tree algorithms were evaluated, as was a softmax regression algorithm that was modified to explicitly reduce the expected Miles-in-Trail cost. The algorithms were evaluated with data from the summer of 2011 for air traffic flows bound to the Newark Liberty International Airport (EWR) over the ARD, PENNS, and SHAFF fixes. The algorithms were provided with 18 input features that describe the weather at EWR, the runway configuration at EWR, the scheduled traffic demand at EWR and the fixes, and other traffic management initiatives in place at EWR. Features describing other traffic management initiatives at EWR and the weather at EWR achieved relatively high information gain scores, indicating that they are the most useful for estimating Miles-in-Trail. In spite of a high variance or over-fitting problem, the decision tree algorithm achieved the lowest expected Miles-in-Trail costs when the algorithms were evaluated using 10-fold cross validation with the summer 2011 data for these air traffic flows.

Bloem, Michael↗

Method and system for dynamic probabilistic risk assessment

The DEFT methodology, system and computer readable medium extends the applicability of the PRA (Probabilistic Risk Assessment) methodology to computer-based systems, by allowing DFT (Dynamic Fault Tree) nodes as pivot nodes in the Event Tree (ET) model. DEFT includes a mathematical model and solution algorithm, supports all common PRA analysis functions and cutsets. Additional capabilities enabled by the DFT include modularization, phased mission analysis, sequence dependencies, and imperfect coverage.

Dugan, Joanne Bechta↗

The Functional Breakdown Structure (FBS) and Its Relationship to Life Cycle Cost

The Functional Breakdown Structure (FBS) is a structured, modular breakdown of every function that must be addressed to perform a generic mission. It is also usable for any subset of the mission. Unlike a Work Breakdown Structure (WBS), the FBS is a function-oriented tree, not a product-oriented tree. The FBS details not products, but operations or activities that should be performed. The FBS is not tied to any particular architectural implementation because it is a listing of the needed functions, not the elements, of the architecture. The FBS for Space Transportation Systems provides a universal hierarchy of required functions, which include ground and space operations as well as infrastructure - it provides total visibility of the entire mission. By approaching the systems engineering problem from the functional view, instead of the element or hardware view, the SPST has created an exhaustive list of potential requirements which the architecture designers can use to evaluate the completeness of their designs. This is a new approach that will provide full accountability of all functions required to perform the planned mission. It serves as a giant check list to be sure that no functions are omitted, especially in the early architectural design phase. A significant characteristic of a FBS is that if architecture options are compared using this approach, then any missing or redundant elements of each option will be ' identified. Consequently, valid Life Cycle Costs (LCC) comparisons can be made. For example, one architecture option might not need a particular function while another option does. One option may have individual elements to perform each of three functions while another option needs only one element to perform the three functions. Once an architecture has been selected, the FBS will serve as a guide in development of the work breakdown structure, provide visibility of those technologies that need to be further developed to perform required functions, and help identify the personnel skills required to develop and operate the architecture. It also wifi allow the systems engineering activities to totally integrate each discipline to the maximum extent possible and optimize at the total system level, thus avoiding optimizing at the element level (stove-piping). In addition, it furnishes a framework that wifi help prevent over or under specifying requirements because all functions are identified and all elements are aligned to functions.

DeHoff, Bryan↗

Automatic Data Filter Customization Using a Genetic Algorithm

This work predicts whether a retrieval algorithm will usefully determine CO2 concentration from an input spectrum of GOSAT (Greenhouse Gases Observing Satellite). This was done to eliminate needless runtime on atmospheric soundings that would never yield useful results. A space of 50 dimensions was examined for predictive power on the final CO2 results. Retrieval algorithms are frequently expensive to run, and wasted effort defeats requirements and expends needless resources. This algorithm could be used to help predict and filter unneeded runs in any computationally expensive regime. Traditional methods such as the Fischer discriminant analysis and decision trees can attempt to predict whether a sounding will be properly processed. However, this work sought to detect a subsection of the dimensional space that can be simply filtered out to eliminate unwanted runs. LDAs (linear discriminant analyses) and other systems examine the entire data and judge a "best fit," giving equal weight to complex and problematic regions as well as simple, clear-cut regions. In this implementation, a genetic space of "left" and "right" thresholds outside of which all data are rejected was defined. These left/right pairs are created for each of the 50 input dimensions. A genetic algorithm then runs through countless potential filter settings using a JPL computer cluster, optimizing the tossed-out data s yield (proper vs. improper run removal) and number of points tossed. This solution is robust to an arbitrary decision boundary within the data and avoids the global optimization problem of whole-dataset fitting using LDA or decision trees. It filters out runs that would not have produced useful CO2 values to save needless computation. This would be an algorithmic preprocessing improvement to any computationally expensive system.

Mandrake, Lukas↗

Forest Canopy Cover and Height from MISR in Topographically Complex Southwestern US Landscape Assessed with High Quality Reference Data

This study addresses the retrieval of spatially contiguous canopy cover and height estimates in southwestern USforests via inversion of a geometric-optical (GO) model against surface bidirectional reflectance factor (BRF) estimates from the Multi-angle Imaging SpectroRadiometer (MISR). Model inversion can provide such maps if good estimates of the background bidirectional reflectance distribution function (BRDF) are available. The study area is in the Sierra National Forest in the Sierra Nevada of California. Tree number density, mean crown radius, and fractional cover reference estimates were obtained via analysis of QuickBird 0.6 m spatial resolution panchromatic imagery usingthe CANopy Analysis with Panchromatic Imagery (CANAPI) algorithm, while RH50, RH75 and RH100 (50, 75, and 100 energy return) height data were obtained from the NASA Laser Vegetation Imaging Sensor (LVIS), a full waveform light detection and ranging (lidar) instrument. These canopy parameters were used to drive a modified version of the simple GO model (SGM), accurately reproducing patterns ofMISR 672 nm band surface reflectance (mean RMSE 0.011, mean R2 0.82, N 1048). Cover and height maps were obtained through model inversion against MISR 672 nm reflectance estimates on a 250 m grid.The free parameters were tree number density and mean crown radius. RMSE values with respect to reference data for the cover and height retrievals were 0.05 and 6.65 m, respectively, with of 0.54 and 0.49. MISR can thus provide maps of forest cover and height in areas of topographic variation although refinements are required to improve retrieval precision.

Chopping, Mark↗

Implementation of a Goal-Based Systems Engineering Process Using the Systems Modeling Language (SysML)

This paper describes the core framework used to implement a Goal-Function Tree (GFT) based systems engineering process using the Systems Modeling Language. It defines a set of principles built upon by the theoretical approach described in the InfoTech 2013 ISHM paper titled "Goal-Function Tree Modeling for Systems Engineering and Fault Management" presented by Dr. Stephen B. Johnson. Using the SysML language, the principles in this paper describe the expansion of the SysML language as a baseline in order to: hierarchically describe a system, describe that system functionally within success space, and allocate detection mechanisms to success functions for system protection.

Breckenridge, Jonathan T.↗

A Semantic Basis for Proof Queries and Transformations

We extend the query language PrQL, designed for inspecting machine representations of proofs, to also allow transformation of proofs. PrQL natively supports hiproofs which express proof structure using hierarchically nested labelled trees, which we claim is a natural way of taming the complexity of huge proofs. Query-driven transformations enable manipulation of this structure, in particular, to transform proofs produced by interactive theorem provers into forms that assist their understanding, or that could be consumed by other tools. In this paper we motivate and define basic transformation operations, using an abstract denotational semantics of hiproofs and queries. This extends our previous semantics for queries based on syntactic tree representations.We define update operations that add and remove sub-proofs, and manipulate the hierarchy to group and ungroup nodes. We show that

Proof Queries↗