Rooted tree representation

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). 

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. 

Fig. 60. Representation of a branched block-copolymer as a full rooted tree left) and as a reduced rooted tree (rightf5 ...

A fourth system of mathematical interest, but of little historical or chemical interest, is the Matula system for naming rooted trees (acyclic alkanes) [10] and its extension to all graphic representation of moieties [11-15], The output of this system is a single very large integer, which can be decoded into a unique acyclic graph. For example, the alkane depicted in Figure 16 has as its Matula name 548,813,133,611. ... 

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

In a parent tree data structure, each successor points to its ancestor. Hence, such a structure can be stored in memory as a sequential list of (node, parent-link) pairs, as illustrated by Fig. 3. The parent tree representation facilitates bottom-up operations, such as finding the (1) root, (2) depth in the tree, and (3) ancestors (i.e., all nodes in the chain from the selected one to the root). Another advantage is in savings in link overhead Only one link per node is required, compared to two per node in the conventional (downward-pointer binary tree) representation. The disadvantage of the parent representation is that it is inefficient for problems requiring either enumeration of all nodes in a tree or top-down exploration of tree sections. Further, it is valid only for nonordered trees. Trees where sibling order is not represented are less versatile data structures. For example, search trees cannot be represented as parent trees, since the search is guided by the order on keys stored in the data structure information fields. 

Fig. 7.3 A representation of the (conjectured) ultrametric distribution of spin-glass equilibrium states. The leaves of the tree at bottom are identified with the states overlaps between states are measured by the number of levels it takes to trace the states back to their common roots . For the three states a, 0 and 7, for example, we have that qot y = q y = q and = 92 > 9l-...

Like checklists, the comprehensiveness of the various predefined trees varies. Some are very detailed with numerous categories and subcategories, whereas others may not fully reach root causes. This is hardly surprising, as the predefined trees are essentially a graphical representation of numerous checklists, organized by subject matter, such as human error, equipment failure, or other topics. The more comprehensive techniques were developed from many years of incident experience and management system experience across the chemical and allied industries. 

Fig. 5. A schematic representation of an m-tree that has m unreacted FUs in the root (R -...

Brooks achieves a representation of classes of objects by an unusual data structure we will call this a "Brooks Data Structure" — BDS. We can adapt the BDS to a LISP-coded specification of a macromolecule as follows the macromolecule is to be described by a TREE, the ROOT of which is the coarsest characterization of the molecule. For a macrocycle, the ROOT would be the great ring for a branched chain it would be the longest backbone. This ROOT would be a list, the basic data structure of LISP, with each element in the list a data set defining a generalized cylinder. 

GDS is a hierarchical tree structure consisting of the root or head node, intermediate nodes, and terminal nodes. Branches connect nodes there may be several branches leaving a node, but only one branch entering a node. The machine representation of GDS provides pointers between these elements and maintains the hierarchy in a form that can be efficiently manipulated. The root of the tree by definition has a (X,Y) coordinate of (0,0) at the origin of the display surface on the device or graphics terminal... 

Artist s IDT algorithm is an adapted version of a Quinlan-like algorithm described in (Bratko, 1990). The result is represented in the form of a Decision Tree (DT), where internal nodes are labelled with attributes, and branches are labelled with attribute values (note, however, that the DT is not the same as the ASS representation discussed earlier). The leaves of the tree are labelled with sound classes. To classify a sound event, a path in the tree is traversed, starting at the root node and ending at a leaf. The IDT algorithm (refer to Appendix 3) proceeds by searching at each non-terminal node for the attribute whose values provide the best discrimination among the other attributes, that is, the Most Informative Attribute (MIA) the formula for the selection of the MIA is explained in Miranda, 1994. 

---