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Definition and Recursive Definition of Binary Trees

– A binary tree is a tree data structure in which each node has at most two children.

– It can be defined as a -ary tree with = 2.

– Binary trees can be interpreted as arborescences in graph theory.

– The recursive definition of a binary tree allows for the distinction between left and right children.

Types of Binary Trees

– Rooted binary tree: has a root node and every node has at most two children.

– Full binary tree: every node has either 0 or 2 children.

– Perfect binary tree: all interior nodes have two children.

– Complete binary tree: every level, except possibly the last, is completely filled.

– Balanced binary tree: the left and right subtrees of every node differ in height by no more than 1.

Properties and Storage of Binary Trees

– The number of nodes in a full binary tree is at least 2h+1 and at most 2^(h+1)-1.

– Binary trees can be used for efficient searching and sorting.

– Binary trees can be stored using nodes and references.

– Special null values or sentinel nodes can be used for nodes with fewer than two children.

– Another storage method is using bits to represent the structure of the tree.

Encoding and Common Operations on Binary Trees

– There is a one-to-one correspondence between ordered trees and binary trees.

– Various operations can be performed on binary trees, such as insertion and deletion.

– Tree rotations are common internal operations on self-balancing binary trees.

– Traversal methods include pre-order, in-order, and post-order.

– Deletion operations help in maintaining the integrity of the binary tree.

Introduction, Types, Operations, and Applications of Binary Trees

– Binary trees are a type of data structure used in computer science.

– Types of binary trees include binary search trees, AVL trees, red-black trees, heap trees, and binary expression trees.

– Operations on binary trees include traversing, searching, inserting, deleting, and balancing.

– Binary trees are used in applications such as file systems, database indexing, expression evaluation, Huffman coding, and dictionary implementations.

Note: The resources and references section is excluded from the groups as it does not contain concepts or detailed points.

Binary tree (Wikipedia)

In computer science, a **binary tree** is a tree data structure in which each node has at most two children, referred to as the *left child* and the *right child*. That is, it is a *k*-ary tree with *k* = 2. A recursive definition using set theory is that a binary tree is a tuple (*L*, *S*, *R*), where *L* and *R* are binary trees or the empty set and *S* is a singleton set containing the root.

From a graph theory perspective, binary trees as defined here are arborescences. A binary tree may thus be also called a **bifurcating arborescence**, a term which appears in some early programming books before the modern computer science terminology prevailed. It is also possible to interpret a binary tree as an undirected, rather than directed graph, in which case a binary tree is an ordered, rooted tree. Some authors use **rooted binary tree** instead of *binary tree* to emphasize the fact that the tree is rooted, but as defined above, a binary tree is always rooted.

In mathematics, what is termed *binary tree* can vary significantly from author to author. Some use the definition commonly used in computer science, but others define it as every non-leaf having exactly two children and don't necessarily label the children as left and right either.

In computing, binary trees can be used in two very different ways:

- First, as a means of accessing nodes based on some value or label associated with each node. Binary trees labelled this way are used to implement binary search trees and binary heaps, and are used for efficient searching and sorting. The designation of non-root nodes as left or right child even when there is only one child present matters in some of these applications, in particular, it is significant in binary search trees. However, the arrangement of particular nodes into the tree is not part of the conceptual information. For example, in a normal binary search tree the placement of nodes depends almost entirely on the order in which they were added, and can be re-arranged (for example by balancing) without changing the meaning.
- Second, as a representation of data with a relevant bifurcating structure. In such cases, the particular arrangement of nodes under and/or to the left or right of other nodes is part of the information (that is, changing it would change the meaning). Common examples occur with Huffman coding and cladograms. The everyday division of documents into chapters, sections, paragraphs, and so on is an analogous example with
*n*-ary rather than binary trees.

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