Binary rooted tree

A decision tree is a binary rooted tree, i.e. there is one initial node v ,. the root, and each node (other than a leaf) has exactly two successors. The leaves are also called terminal nodes, all remaining nodes are internal nodes. Internal nodes beeu decision rules of the form Xj < a, terminal nodes bear function values %. The node numbering is such that an internal node has successors V2fc+i and V2fc+2> see Figure 6.4. 

This complicated triple sum can be simplified by applying the rooted tree treatment. Here the outline is confined to a binary copolymer the results may easily be generalized to copolymers with r components. 

To keep the description of the regions simple, binary decision trees with a root, nodes, and leaves, the R, s, are favored. Each region is split into two further regions over and over again termed a recursive binary partitioning. The decision for the binary split is based on a constant, s, as shown in Figure 5.34b. 

In a Fibonaccian search, the elements of the binary search tree are either Fibonacci numbers or derived from them the root is a Fibonacci number, as are all nodes reached by only left links. Right links lead to nodes whose values are the ancestor plus the difference between it and its left successor. That is, the difference between the ancestor and left successor is added to the ancestor to get the right successor value. Fibonaccian binary search trees have a total number of elements one less than a Fibonacci number. 

One way of thinking of this situation is to visualize a game, with a binary tree of depth N as a "board" and k markers or "pieces" (representing locations in a program scheme). Any marker can be placed on a leaf. Each node is either a leaf or has two "sons". If both sons of a node are covered with markers, then one marker can be moved up to that node and markers removed from the two sons. The object is to eventually place a marker on the root, using as few markers as possible. How many markers are needed in the worst case ... 

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... 

The linear programming LP relaxation of the MILP model is the most frequently used type of relaxation in branch and bound algorithms. In the root node of a binary tree, the LP relaxation of the MILP model of (1) takes the form ... 

In both schemes the subsets and the partitions are obtained by imagining the receivers as the leaves in a rooted full binary tree with N leaves (assume that iV is a power of 2). Such a tree contains 2N — 1 nodes (leaves plus internal nodes) and for any 1 < i < 2N — 1 we assume that Uj is a node in the tree. The systems differ in the collections of subsets they consider. 

The collection of subsets Si,..., S-uj in our first scheme corresponds to all complete subtrees in the full binary tree with N leaves. For any node Vi in the full binary tree (either an internal node or a leaf, 2N — 1 altogether) let the subset Si be the collection of receivers u that correspond to the leaves of the subtree rooted at node -y,. In other words, u E Si iff Vi is an ancestor of u. The key assignment method is simple assign an independent and random key Li to every node Vi in the complete tree. Provide every receiver u with the log AT -I-1 keys associated with the nodes along the path from the root to leaf u. 

For a given set H of revoked receivers, let ui,... be the leaves corresponding to the elements in IZ. The method to partition M TZ into disjoint subsets is as follows. Consider the (directed) Steiner Tree ST TZ) defined by the set 71 of vertices and the root, i.e. the minimal subtree of the full binary tree that connects all the leaves in TZ. ST 7Z) is unique. Let Si, ..., Si be all the subtrees of the original tree that hang off ST 7Z), that is, all subtrees whose roots ui,..., are adjacent to nodes of outdegree 1 in ST 7Z), but they are not in ST 7Z). The next claim follows immediately and shows that this construction is indeed a cover, as required. 

Main key generation On input the parameters and a new prekey prek, N one-time key pairs (sk tempj, mkp based on prek are generated. The values mkj are used as the leaves of a binary tree. The value of each inner node is the hash value of its two children (regarded as one message). The new temporary secret key sk temp consists of the whole tree and a counter for the messages already signed, initialized with zero. But only the final hash value, i.e., the root of the tree, is published as the main public key mk of the new scheme. 

Signing (sign ) Signing takes place in a complete binary tree with N leaves. Only the root exists at the beginning the other nodes are created on demand. Each node is denoted by a bit string Z, where the root is Node e and the children of Node I are Nodes 10 and Z1 (for each / of length < d, where N = 2 ). 

A heap is a size-ordered complete binary tree. The root of the tree is thus either the largest of the key values or the least, depending on the convention adopted. When a heap is built, a new key is inserted at the first free node of the bottom level (just to the right of the last filled node), then exchanges take place (bubbling) until the new value is in the place where it belongs. 

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. 

Construction operations can be represented in a binary tree structure called a CSG tree. The part in Figure 5-5a is created by a sequence of combination operations according to Figure 5-5b. A binary tree (Figure 5-5c) represents the final shape by its root node. 

Builds a binary tree, placing operators into the tree according to their degree of shared variable connectivity, with more connected operators closer to the root of the tree. Bottom-up tree traversal algorithms are used to cluster operators and values and perform fiinctional unit allocation and register allocation, respectively. A top-down tree traversal algorithm is used to assign interconnect. 

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