Tree representation

Tree representation of the conformation search problem for hexane. Unlike the tree in Figure 9.4 the path gth from the root node to any of the terminal nodes is constant. 

An analysis is conducted of the predicted values for each team member s factorial table to determine the main effects and interactions that would result if the predicted values were real data The interpretations of main effects and interactions in this setting are explained in simple computational terms by the statistician In addition, each team member s results are represented in the form of a hierarchical tree so that further relationships among the test variables and the dependent variable can be graphically Illustrated The team statistician then discusses the statistical analysis and the hierarchical tree representation with each team scientist ... 

Fig. 5. Two typical rooted tree representations of a four ray star-molecule. Tb the branch point selected as root Ttj the j-th element of a ray selected as root...

Fig. 6 a, b. A tetrafunctionally branched molecule (a) placed on a lattice and (b) the corresponding rooted tree representation. Note The units in the first, second, third etc. shell of neighbours come to lie well defined in generation gt, g2, g3 etc... 

Second, on placing a molecule on a special lattice, a picture is unconsciously engraved in the mind suggesting that the molecule may behave in three-dimensional space as seen in the graph or given by the computer. A special lattice always implies certain constraints which actually need not exist in this form. The rooted tree representation is free from this problem of how a molecule is embedded in space it only displays the connectivity, and this in a very clear form95 97). 

Fig. 16. Example of tree-like representation for RNA secondary structure. Each hairpin structure is shown next to its equivalent tree. With such representations, a graph theoretic measure can measure the distance between these trees and help generate fitness values for a fitness landscape. For example, the distance between two structures may be defined as the minimal number of elementary graph operations (insert a point, switch an edge, etc.) needed to convert one tree into the other. Note that there are many variants of tree representations for RNA secondary structures and many definitions of graph distance. In low-resolution tree representations, several secondary structures can map to the same graph.

A workspace by itself has a flat data structure with a minimum of predefined hierarchy. The data organization can be structured on the fly by using the metadata of each file. Since each file may have several metadata that can be used for categorization, the user defines the file tree representation within each session and each workspace independently for other views. Different user-specific tree view settings can then be stored and selected for later use. 

Structurality. The structurality has been introduced [Duchateau 2009] to intuitively measure the qualities of the structure an object possesses.6 In the case of schemas, this notion is translated to the set of ancestors of a schema structure. In other words, the structurality measures whether the elements of the generated and the intended schema contain the same set of ancestors. To compute structurality, the schemas are viewed as trees. Let S r and Sgen denote the intended and the generated target schema, respectively. Assume also that in the tree representation of a schema S, Ps (e) is the set of elements in the path from the root to the element e, exclusively. The structurality of an element e is defined as follows ... 

Figure 2a. SMILES string tree representation for decahydro-2,3-dimethylnaphthalene, C(C(CCC1)CC(C2C))(C1)C2.

Figure 2b. SMILES string tree representation for SMILES string tree from a with Cl as root.

Figure 17 Tree representation of a systematic conformational search (grid search).

T o enhance the performance of grid searches, two advantages of the tree representation can be utilized. Firstly, parts of the tree that correspond to conformations with atom clashes or close contacts can be detected and pruned. Fig. 18 illustrates this for a conformation of n-heptane. Neither changes to the torsion xn nor to rn, will avoid the steric clash exhibited at the marked atoms. Thus, the search tree can be pruned above the node of xn i-... 

A tree representation is helpful in understanding the systematic search and how the basic algorithm described above can be improved. Trees are frequently... 

Figure 3 A tree representation of the conformational space of a molecule in which the first dihedral angle can adopt three values, the second dihedral can adopt two values, and the third dihedral can adopt three values. There are a total of 18 possible conformations.

P.-N. Guo, C.-K Cheng, T. Yoshimura. An O-tree representation of non-slicing floorplan and its applications. In Proc. Design Automation Conf., 1999. 

More efficient is a generic tree representation of the separations based on tasks. The sequencing can be formulated as a structural optimisation problem where standard techniques based on Mixed Integer Linear Programming (MILP) apply. The tasks consist of simple distillation columns, as well as of hybrids for complex column arrangements, modelled by appropriate shortcut or semi-rigorous methods. Details can be found in Doherty and Malone (2001). 

Visualization is an efficient way to utilize the human ability to process laige amounts of data. Traditional visualization methods are based on clustering and tree representation, and are complemented by projecting objects onto a Euclidean space to reflect theh stractural or functional differences. The data are visualized without preclustering and can be explored dynamically and interactively, e.g., in protein topology and gene expression. 

Whilst the 2D-DWT provides an efficient space-frequency characterisation of a given image, it only uses a fixed decomposition of the pixel space. As in the case of the 1-D wavelet packet transform, we can extend the wavelet packets to two dimensions. That is, the 2D wavelet packet transform (2D-WPT) generates a more general, full m -ary tree representation with a total of m + m + m sub-bands for h levels. Each sub-band in a given level of the tree splits into a smoothed sub-band and m"-l detailed sub-bands, resulting in a tree that resembles an m-way pyramidal stack of sub-bands. For the case of a dyadic decomposition scheme, this corresponds to a pyramidal sub-band structure where each sub-band is decomposed into 2 = 4 sub-bands at each successive (higher) level (see Fig. 2). Fig. 7 shows results of the third level of the 2D WPT for the dyadic case - a total of = 64... 

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