Subtree

It is possible to represent molecules with feature trees at various levels of resolution. The maximum simplification of a molecule is its representation as a feature tree with a single node. On the other hand, each acyclic atom forms a node at the highest level. Due to the hierarchical nature of feature trees, all levels of resolution can be derived from the highest level. A subtree is replaced by a single node which represents the union of the atom sets of the nodes belonging to this subtree. 

A fault tree is a graphical form of a Boolean equation, but the probability of the top event (and lesser events) can be found by substituting failure rates and probabilities for these iwo-staie events. The graphical fault tree is prepared for computer or manual evaluation by pruning" it of less significant events to focus on more significant events. Even pruned, the tree may be so large that it IS intractable and needs division into subtrees for separate evaluations. If this is done, care must be taken to insure that no information is lost such as interconnections between subtrees. 

FTAP < characier " -ic nameSs rmation, scripiion AND K-of N NOT/ le computer meniory is limiting factor Minimal cut of up to 10 can be generated top-down, bo and Nelson memuu prime implicants f - i cut sets and [ji mic implicants Independent subtrees ibiind and replaced by inodijle IBM C sm 1IV, Available from Ope Rest Uni Cali ley... 

Choose this to build and edit-event trees. The Event Tree List dialog appears listing all event trees and subtrees in the current family. A pop-up menu provides the following options ... 

Fig. 5.3.1 Fault Tree Identification of Flood-Ci Equipmentithe elevation information enters in the subtrees...

Accident progression scenarios are developed and modeled as event trees for each of these accident classes. System fault trees are developed to the component level for each branch point, and the plant response to the failure is identified. Generic subtrees are linked to the system fault trees. An example is "loss of clcciric power" which is analyzed in a Markov model that considers the frequencies of lo,sing normal power, the probabilities of failure of emergency power, and the mean times to repair parts of the electric power supply. 

A decision tree for Design Methodology is illustrated in Fig. 3.2. Each step in the tree is explained briefly below. The steps have also their own subtrees, which are described separately. 

Select the optimal ventilation method to reduce emissions and/or to pro tect workers. (For details, see the subtree Fig.. 3.10.)... 

FIGURE 5.10. OfEshore drilling blowout fault tree subtree. 

FIGURE 5.10. Offshore Drilling Blowout Fault Tree Subtree, "Fail to use shear rams to prevent blowout." 221... 

Consider, for definiteness, a set of otherwise identical lowest-level components of a system, so that the hierarchy is a tree of constant depth. Since we assume that the components are all identical, the only distinction among the various nodes of the hierarchy consists of the structure of the subtrees. Now suppose we have a tree T that consists of /3 subtrees branching out from the root at the top level. We need to determine the number of different interactions that can occur on each level, independent of the structure of each subtree i.e. isomorphic copies of trees do not contribute to our count. We therefore need to find the number of nonisomorphic subtrees. We can do this recursively. 

The diversity of the tree T, denoted by V(T), counts the total number of interactions between and within all subtrees. We therefore proceed in two steps. First count the number of distinct interactions within the clusters represented by the subtrees i.e. multiply the diversity of all nonisomorphic subtrees. Second, multiply this result by the number of ways, N, that k different clusters... 

Fig. 12.5 Examples of trees whose subtrees are all identical (a) and different (b).

For example, consider the two trees shown in figure 12.4. The tree in figure 12.4-a has equal (binary) subtrees, so that its diversity V = 1 and complexity C = 0. The same is true for any tree all of whose nodes have a constant branching ratio. On the other hand, the tree shown in figure 12.4-b has two distinct subtrees, each of which has a diversity of 2 = 1. The diversity of the entire tree is therefore X) = 22 - 1 = 3. 

Utilizing only the simple lower bounding scheme with the bottleneck machine fixed to the last machine, we generate the branching tree given in Fig. 3, where much of the detail has been suppressed. The boldface numbers indicate the size of the subtrees (i.e., the number of nodes in the subtree) beneath the node. 

The last step in the preceding argument, the use of our knowledge about flowshop scheduling, turns what had been a mainly syntactic criterion over the tree structure of the example, into a criterion based on state variables of (jc, y). The state variable values, the completion times of the various flowshop machines, are accessible before the subtrees beneath jc and y have been generated. Indeed, they determine the relationships between the respective elements of the subtrees (jcm, yu). If we can formalize the process of showing that the pair (jc, y) identified with our syntactic criterion, satisfies the eonditions for equivalence or dominance, wc will in the process have generated a new equivalence rule. 

The analysis of problem-solving experience that has taken place so far has been based on finding subproblems within an existing branching structure that, when solved, will produce subtrees that satisfy the definitions of dominance and equivalence. As we have noted, this is insufficient for generating new dominance and equivalence conditions because we... 

J.B. Kruskal Jr., On the shortest spanning subtree of a graph and the traveling salesman problem. Proc. Am. Math. Soc., 7 (1956) 48-50. 

The synthesis is carried out in two phases. In the first phase a heuristic starting routine is used to generate an initial tree. It was found through experience that the pipes connected to the separation plant (the root of the tree) play a special role in the development of a tree. These pipes will be referred to as arms and the pipes connected to each arm its subtree. It is assumed that the number and location of the arms are specified as program input by the engineer. The two heuristic rules used by the starting routine are (i) efficient trees have low total pipe mileage (ii) efficient trees have nearly equal flow in their arms. Notice that the application of rule (i) by itself... 

To make this proof constructive, the names of nodes should form a recursive set, there should be a total recursive function f such that f(n) lists all the sons of node n, and the predicate "The subtree rooted at n is infinite" should be total recursive. 

The answer is that if the tree is a full binary tree of depth N, then N+l markers are necessary and sufficient. The proof is by induction on N. The case N - 1 is obvious, for then there are just two leaves and both must be covered before the root can be covered. Suppose this is true for depth N-l. The root has two sons each of which can be regarded as the root of a full binary subtree of depth N-l call these nodes n and n. The root can be covered when and only... 

Beyond the gel point, the bonds Issuing from a monomer unit can have finite or Infinite continuation. If the continuation Is finite, the Issuing subtree Is also only finite If the continuation Is Infinite, the unit Is bound via this bond to the "infinite" gel. The classification of bonds with respect to whether they have finite or Infinite continuation enables a relatively detailed statistical description of the gel structure. The probability of finite continuation of a bond Is called the extinction probability. The extinction probability Is obtained In a simple way from the distribution of units In generation g>0. This distribution Is obtained from the distribution of units In the root g-0 (for more details see Ref. 6). 

Each vertex v except the root has a unique parent u which is adjacent to v and satisfies h(u) > h(v)+l. All other vertices w adjacent to v are called its children and satisfy h(w) < h(v)-l. We define ancestors and descendants in the obvious way. Each vertex v in the tree defines a subtree consisting of v and its descendants (see Figure 6). 

That is, to compare two trees, we first compare their root labels. If these are identical, we order the subtrees defined by the children of the roots, and compare the ordered sequences of subtrees lexicographically. 

On trees, not only is the isomorphism problem efficiently solvable, but so is the subgraph isomorphism problem. Edmonds and Matula (29) have discovered an algorithm which will determine whether one tree is isomorphic to a subtree of another in... 

Figure 6. Tree of Figure 5 in canonical form. Dashes enclose subtrees of children of the root. 

These functions 0 represent the scattering behavior of a full subtree that emerges from a functional group of a monomer selected at random. For instance ... 

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