Binary Tree
A binary tree is a data structure composed of nodes, where each node has at most two children, referred to as the left child and the right child. The topmost node is called the root. Binary trees are commonly used in computer science for efficient searching and sorting algorithms, and they provide a foundation for more complex data structures like binary search trees and AVL trees.
3 Key excerpts on "Binary Tree"
eBook - ePubMesh Generation
Application to Finite Elements
- Pascal Frey, Paul Louis George(Authors)
- 2013(Publication Date)
- Wiley-ISTE(Publisher)
...We noticed the running time improvement over the naive algorithm of Sections 2.3. In this section, we do the same in a dynamic context based on a Binary Search Tree (referred to hereafter as BST) that improves the complexity of any searching operation on linked lists. We start with a definition. Figure 2.8. Pointer representation of a Binary Tree. Definition 2.1 A Binary Tree is a data structure whose node contains, in addition to the information field (the value), two pointers; the left and the right children. If the information field obeys some ordering relationship, such a tree is called a Binary Search Tree (BST). The topmost node is called the root of the tree. A distinction is made between a node that has children, called internal, and a node without children, called terminal or a leaf The depth (height) of a node is the number of edges (branches) crossed from the root to that node (Figure 2.8). Figure 2.9. Binary Tree constructed from the sequence of values 17, 5, 1 and 3. Notice that in this tree, according to the sequence of the values, the nodes have only one child. A different ordering could lead to nodes having two children. An example of binary search tree growth is illustrated in Figure 2.9 where the insertion of the values 17, 5, 1 and 3 is depicted. First, 17 is inserted and stored at the root since the tree is empty. Then 5 is inserted and put in the left subtree as it is smaller than 17. Similarly, 1 goes to the left of 5 and 3 to the right of 1. The resulting tree has one internal node at depths 0, 1, 2 and a leaf 3, it has an edge at depths 1, 2 and 3. To see which parameters influence the operations on BST, let us consider the searching and insertion procedures whose general schemes are given below: Algorithm 2.10 Searching in a BST. Algorithm 2.11 Inserting a value in a BST. Basically, the strategy consists of tracing a path down the tree and making the right decision at each node encountered...
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A Language for Learning
- Michael Friendly(Author)
- 2014(Publication Date)
- Psychology Press(Publisher)
...The basic form of a Binary Tree as a list is this: A Binary Tree is a list containing three items: [ NODE LEFT.BRANCH RIGHT.BRANCH ] which you can draw like this: or this: When these trees are shown growing horizontally, as in the first diagram above, left and right in the diagram really corresponds to various depths in the tree. Nevertheless, I will continue to describe these trees in terms of left and right and use the name “left branch” to refer to the branch above the node. Here is where we get tricky: Either the left branch or the right branch (or both) may also be Binary Trees, that is, a list of three elements: [ NODE [ NODE LEFT.BRANCH RIGHT.BRANCH ] [ NODE LEFT.BRANCH RIGHT.BRANCH ] ] For example, Figure 8–9 shows a partial pedigree tree for Charles, Prince of Wales. The first two generations can be represented by this list: [ CHARLES [ELIZABETH.II ELIZABETH GEORGE.VI] [PHILIP ALICE ANDREW ] ] Fig. 8–9 Pedigree tree of Charles, Prince of Wales Again, the word at either the left or the right branch (or both) can be replaced by a three-item list to extend the tree to one more generation. (As the figure shows, it is not necessary for the tree to extend to the same depth at all branches)...
eBook - ePub- Stephen Wise(Author)
- 2018(Publication Date)
- CRC Press(Publisher)
...In Figure 12.20, the root node has two values which, as well as storing data, or pointers to data, act as indices to the child nodes in much the same way as a Binary Tree. The left child contains values which are all smaller than A, the middle child values between A and B and the right child values which are greater than B. Each child node in this case has 10 entries, but this is simply to make it possible to produce a diagram. In practice, nodes in a B tree are designed to be large enough that they fill a significant portion of a disk block and may have of the order of 100 entries. What is more, each node can also have child nodes. For example, the entries in the left node of Figure 12.20 would have 10 values in numerical order, and these would act as indices to 11 child nodes, each with 10 entries. The root node in Figure 12.20 only has two entries, but it could also have 10 entries, and thus 11 children. This B-tree therefore has space to hold over 1000 data values in a tree which is only three levels deep. FIGURE 12.20 Structure of a B-tree. The price which is paid with a B tree is that when a node is returned, you know that it will contain the value you are looking for, but you will need to do a linear search through the values in the node to find it. However, because disk access is so slow, searching N values in memory is much faster than doing the log(N) disk accesses it could take to find the same value in a Binary Tree which was stored on disk. To understand how the B-tree might work to store our Morton codes, we will use a much smaller tree as shown in Figure 12.21. This is known as a 2–3 B-tree because each node has up to two values and can have up to three children. To understand how a B-tree works, it is helpful to consider how the tree in Figure 12.21 was built. There are various ways to build a B-tree, one of which is to start with an empty tree and add the data values one at a time...
