TheInfoList

In
computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of , , and . Computer science ...
, a binary tree is a
tree data structure In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of , ...
in which each node has at most two
children Biologically, a child (plural children) is a being between the stages of and , or between the of and puberty. The legal definition of ''child'' generally refers to a , otherwise known as a person younger than the . Children generally have ...
, which are referred to as the ' and the '. A
recursive definition In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
using just
set theory Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that are relevant to ...
notions is that a (non-empty) binary tree is a
tuple In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
(''L'', ''S'', ''R''), where ''L'' and ''R'' are binary trees or the
empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, cha ... and ''S'' is a
singleton set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
containing the root. Some authors allow the binary tree to be the empty set as well. From a
graph theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
perspective, binary (and K-ary) trees as defined here are arborescences. A binary tree may thus be also called a bifurcating arborescence—a term which appears in some very old programming books, before the modern computer science terminology prevailed. It is also possible to interpret a binary tree as an
undirected In mathematics, and more specifically in graph theory, a graph is a structure amounting to a Set (mathematics), set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions cal ... , rather than a
directed graph In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a Graph (discrete mathematics), graph that is made up of a set of Vertex (graph theory), vertices connected by directed Edge (graph theory), edges often called ... , in which case a binary tree is an ordered,
rooted tree In graph theory, a tree is an undirected graph in which any two Vertex (graph theory), vertices are connected by ''exactly one'' Path (graph theory), path, or equivalently a Connected graph, connected Cycle (graph theory), acyclic undirected graph ... . 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. A binary tree is a special case of an ordered
K-ary tree In graph theory, an ''m''-ary tree (also known as ''k''-ary or ''k''-way tree) is a rooted tree (graph theory), tree in which each node has no more than ''m'' children. A binary tree is the special case where ''m = 2'', and a ternary tree is ...
, where ''k'' is 2. 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 order (as left/right) the children either. In computing, binary trees are 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 tree In computer science, a binary search tree (BST), also called an ordered or sorted binary tree, is a Rooted tree, rooted binary tree whose internal nodes each store a key greater than all the keys in the node's left subtree and less than those in ... s and
binary heap A binary heap is a heap (data structure), heap data structure that takes the form of a binary tree. Binary heaps are a common way of implementing priority queues. The binary heap was introduced by J. W. J. Williams in 1964, as a data structure for ... s, and are used for efficient
searching Searching or search may refer to: Computing technology * Search algorithm, including keyword search ** :Search algorithms * Search and optimization for problem solving in artificial intelligence * Search engine technology, software for findin ...
and
sorting Sorting is any process of arranging items systematically, and has two common, yet distinct meanings: # ordering Order or ORDER or Orders may refer to: * Orderliness, a desire for organization * Categorization, the process in which ideas and o ...
. 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 In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of Algor ...
and
cladograms A cladogram (from Greek language, Greek ''clados'' "branch" and ''gramma'' "character") is a diagram used in cladistics to show relations among organisms. A cladogram is not, however, an Phylogenetic tree, evolutionary tree because it does not s ...
. The everyday division of documents into chapters, sections, paragraphs, and so on is an analogous example with n-ary rather than binary trees.

# Definitions

## Recursive definition

To actually define a binary tree in general, we must allow for the possibility that only one of the children may be empty. An artifact, which in some textbooks is called an ''extended binary tree'' is needed for that purpose. An extended binary tree is thus recursively defined as: * the
empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, cha ... is an extended binary tree * if T1 and T2 are extended binary trees, then denote by T1 • T2 the extended binary tree obtained by by adding edges when these sub-trees are non-empty. Another way of imagining this construction (and understanding the terminology) is to consider instead of the empty set a different type of node—for instance square nodes if the regular ones are circles.

## Using graph theory concepts

A binary tree is a
rooted tree In graph theory, a tree is an undirected graph in which any two Vertex (graph theory), vertices are connected by ''exactly one'' Path (graph theory), path, or equivalently a Connected graph, connected Cycle (graph theory), acyclic undirected graph ... that is also an
ordered tree In graph theory In mathematics, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph the ...
(a.k.a. plane tree) in which every node has at most two children. A rooted tree naturally imparts a notion of levels (distance from the root), thus for every node a notion of children may be defined as the nodes connected to it a level below. Ordering of these children (e.g., by drawing them on a plane) makes it possible to distinguish a left child from a right child. But this still doesn't distinguish between a node with left but not a right child from a one with right but no left child. The necessary distinction can be made by first partitioning the edges, i.e., defining the binary tree as triplet (V, E1, E2), where (V, E1 ∪ E2) is a rooted tree (equivalently arborescence) and E1 ∩ E2 is empty, and also requiring that for all ''j'' ∈ every node has at most one E''j'' child. A more informal way of making the distinction is to say, quoting the
Encyclopedia of Mathematics The ''Encyclopedia of Mathematics'' (also ''EOM'' and formerly ''Encyclopaedia of Mathematics'') is a large reference work in mathematics. Overview The 2002 version contains more than 8,000 entries covering most areas of mathematics at a graduate ...
, that "every node has a left child, a right child, neither, or both" and to specify that these "are all different" binary trees. also in print as

# Types of binary trees

Tree terminology is not well-standardized and so varies in the literature. * A binary
tree In botany, a tree is a perennial plant with an elongated Plant stem, stem, or trunk (botany), trunk, supporting branches and leaves in most species. In some usages, the definition of a tree may be narrower, including only wood plants with se ...
has a
root node In computer science, a tree is a widely used abstract data type that simulates a hierarchical tree structure, with a root value and subtrees of children with a #Terminology, parent node, represented as a set of linked Node (computer science), nod ...
and every node has at most two children.  * A binary tree (sometimes referred to as a proper or plane binary tree) is a tree in which every node has either 0 or 2 children. Another way of defining a full binary tree is a
recursive definition In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. A full binary tree is either: ** A single vertex. ** A tree whose root node has two subtrees, both of which are full binary trees. * In a binary tree every level, ''except possibly the last'', is completely filled, and all nodes in the last level are as far left as possible. It can have between 1 and 2''h'' nodes at the last level ''h''. An alternative definition is a perfect tree whose rightmost leaves (perhaps all) have been removed. Some authors use the term complete to refer instead to a perfect binary tree as defined below, in which case they call this type of tree (with a possibly not filled last level) an almost complete binary tree or nearly complete binary tree. A complete binary tree can be efficiently represented using an array. * A binary tree is a binary tree in which all interior nodes have two children ''and'' all leaves have the same ''depth'' or same ''level''. An example of a perfect binary tree is the (non-incestuous)
ancestry chart An ancestor, also known as a forefather, fore-elder or a forebear, is a parent A parent is a caregiver of the offspring in their own species. In humans, a parent is the caretaker of a child (where "child" refers to offspring, not necessarily ...
of a person to a given depth, as each person has exactly two biological parents (one mother and one father). Provided the ancestry chart always displays the mother and the father on the same side for a given node, their sex can be seen as an analogy of left and right children, ''children'' being understood here as an algorithmic term. A perfect tree is therefore always complete but a complete tree is not necessarily perfect. * In the infinite complete binary tree, every node has two children (and so the set of levels is
countably infinite In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
). The set of all nodes is countably infinite, but the set of all infinite paths from the root is uncountable, having the
cardinality of the continuum In set theory illustrating the intersection of two sets Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, ...
. That's because these paths correspond by an order-preserving
bijection In , a bijection, bijective function, one-to-one correspondence, or invertible function, is a between the elements of two , where each element of one set is paired with exactly one element of the other set, and each element of the other set is p ... to the points of the
Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883 ... , or (using the example of a Stern–Brocot tree) to the set of positive
irrational number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s. * A balanced binary tree is a binary tree structure in which the left and right subtrees of every node differ in height by no more than 1. One may also consider binary trees where no leaf is much farther away from the root than any other leaf. (Different balancing schemes allow different definitions of "much farther".) * A degenerate (or pathological) tree is where each parent node has only one associated child node. This means that the tree will behave like a
linked list In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of , ... data structure.

# Properties of binary trees

* The number of nodes $n$ in a full binary tree, is at least $n = 2h+1$ and at most $n = 2^-1$, where $h$ is the
height 200px, A cuboid demonstrating the dimensions length, width">length.html" ;"title="cuboid demonstrating the dimensions length">cuboid demonstrating the dimensions length, width, and height. Height is measure of vertical distance, either vertical ... of the tree. A tree consisting of only a root node has a height of 0. * The number of leaf nodes $l$ in a perfect binary tree, is $l = \left(n + 1\right) / 2$ because the number of non-leaf (a.k.a. internal) nodes $n - l = \sum_^ 2^k = 2^ - 1 = l - 1$. * This means that a full binary tree with $l$ leaves has $n = 2l - 1$ nodes. * In a balanced full binary tree, $h = \lceil \log_2\left(l\right)\rceil + 1 = \lceil \log_2\left(\left(n + 1\right) / 2\right)\rceil + 1 = \lceil \log_2\left(n + 1\right)\rceil$ (see
ceiling function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
). * In a perfect full binary tree, $l = 2^$ thus $n = 2^ - 1$. * The number of null links (i.e., absent children of the nodes) in a binary tree of ''n'' nodes is (''n''+1). * The number of internal nodes in a complete binary tree of ''n'' nodes is $\lfloor n/2\rfloor$. * For any non-empty binary tree with ''n''0 leaf nodes and ''n''2 nodes of degree 2, ''n''0 = ''n''2 + 1.

# Combinatorics

In
combinatorics Combinatorics is an area of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geom ...
one considers the problem of counting the number of full binary trees of a given size. Here the trees have no values attached to their nodes (this would just multiply the number of possible trees by an easily determined factor), and trees are distinguished only by their structure; however, the left and right child of any node are distinguished (if they are different trees, then interchanging them will produce a tree distinct from the original one). The size of the tree is taken to be the number ''n'' of internal nodes (those with two children); the other nodes are leaf nodes and there are of them. The number of such binary trees of size ''n'' is equal to the number of ways of fully parenthesizing a string of symbols (representing leaves) separated by ''n'' binary operators (representing internal nodes), to determine the argument subexpressions of each operator. For instance for one has to parenthesize a string like , which is possible in five ways: : $\left(\left(X*X\right)*X\right)*X,\qquad \left(X*\left(X*X\right)\right)*X,\qquad \left(X*X\right)*\left(X*X\right),\qquad X*\left(\left(X*X\right)*X\right),\qquad X*\left(X*\left(X*X\right)\right).$ The correspondence to binary trees should be obvious, and the addition of redundant parentheses (around an already parenthesized expression or around the full expression) is disallowed (or at least not counted as producing a new possibility). There is a unique binary tree of size 0 (consisting of a single leaf), and any other binary tree is characterized by the pair of its left and right children; if these have sizes ''i'' and ''j'' respectively, the full tree has size . Therefore, the number $C_n$ of binary trees of size ''n'' has the following recursive description $C_0=1$, and $\textstyle C_n=\sum_^C_iC_$ for any positive integer ''n''. It follows that $C_n$ is the
Catalan number In combinatorial mathematics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite set, finite Mathematical structure, structures. It is ... of index ''n''. The above parenthesized strings should not be confused with the set of words of length 2''n'' in the
Dyck languageDyck is a form the Dutch surname (van) Dijck, which is also wide spread among Russian Mennonites. It may refer to: * Anthony van Dyck (1599–1641), Flemish artist * Cornelius Van Allen Van Dyck (1818–1895), American missionary * Howard Dyck (bor ...
, which consist only of parentheses in such a way that they are properly balanced. The number of such strings satisfies the same recursive description (each Dyck word of length 2''n'' is determined by the Dyck subword enclosed by the initial '(' and its matching ')' together with the Dyck subword remaining after that closing parenthesis, whose lengths 2''i'' and 2''j'' satisfy ); this number is therefore also the Catalan number $C_n$. So there are also five Dyck words of length 6: : $\left(\right)\left(\right)\left(\right),\qquad \left(\right)\left(\left(\right)\right),\qquad \left(\left(\right)\right)\left(\right),\qquad \left(\left(\right)\left(\right)\right),\qquad \left(\left(\left(\right)\right)\right)$. These Dyck words do not correspond to binary trees in the same way. Instead, they are related by the following recursively defined bijection: the Dyck word equal to the empty string corresponds to the binary tree of size 0 with only one leaf. Any other Dyck word can be written as ($w_1$)$w_2$, where $w_1$,$w_2$ are themselves (possibly empty) Dyck words and where the two written parentheses are matched. The bijection is then defined by letting the words $w_1$ and $w_2$ correspond to the binary trees that are the left and right children of the root. A bijective correspondence can also be defined as follows: enclose the Dyck word in an extra pair of parentheses, so that the result can be interpreted as a
Lisp Lisp (historically LISP) is a family of programming language A programming language is a formal language In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbo ...
list expression (with the empty list () as only occurring atom); then the dotted-pair expression for that proper list is a fully parenthesized expression (with NIL as symbol and '.' as operator) describing the corresponding binary tree (which is, in fact, the internal representation of the proper list). The ability to represent binary trees as strings of symbols and parentheses implies that binary trees can represent the elements of a
free magma 300px, Algebraic structures between magmas and groups. In abstract algebra, a magma, binar or groupoid is a basic kind of algebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...
on a singleton set.

# Methods for storing binary trees

Binary trees can be constructed from
programming language A programming language is a formal language In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (computer science) ... primitives in several ways.

## Nodes and references

In a language with
records A record, recording or records may refer to: An item or collection of data Computing * Record (computer science) In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures ...
and
references Reference is a relationship between objects in which one object designates, or acts as a means by which to connect to or link to, another object. The first object in this relation is said to ''refer to'' the second object. It is called a ''name'' ...
, binary trees are typically constructed by having a tree node structure which contains some data and references to its left child and its right child. Sometimes it also contains a reference to its unique parent. If a node has fewer than two children, some of the child pointers may be set to a special null value, or to a special
sentinel node In computer programming, a sentinel node is a specifically designated node (computer science), node used with linked lists and Tree (data structure), trees as a traversal path terminator. This type of node does not hold or reference any data man ...
. This method of storing binary trees wastes a fair bit of memory, as the pointers will be null (or point to the sentinel) more than half the time; a more conservative representation alternative is threaded binary tree. In languages with
tagged unionIn computer science, a tagged union, also called a variant type, variant, variant record, choice type, discriminated union, disjoint union, sum type or coproduct, is a data structure used to hold a value that could take on several different, but fixe ...
s such as ML, a tree node is often a tagged union of two types of nodes, one of which is a 3-tuple of data, left child, and right child, and the other of which is a "leaf" node, which contains no data and functions much like the null value in a language with pointers. For example, the following line of code in
OCaml OCaml ( , formerly Objective Caml) is a general-purpose, multi-paradigm programming language Programming paradigms are a way to classify programming languages based on their features. Languages can be classified into multiple paradigms. S ...
(an ML dialect) defines a binary tree that stores a character in each node. type chr_tree = Empty , Node of char * chr_tree * chr_tree

## Arrays

Binary trees can also be stored in breadth-first order as an
implicit data structureIn computer science, an implicit data structure or space-efficient data structure is a data structure that stores very little information other than the main or required data: a data structure that requires low Overhead (computing), overhead. They ar ...
in
arrays ARRAY, also known as ARRAY Now, is an independent distribution company launched by film maker and former publicist Ava DuVernay Ava Marie DuVernay (; born August 24, 1972) is an American filmmaker. She won the directing award in the U.S. drama ...
, and if the tree is a complete binary tree, this method wastes no space. In this compact arrangement, if a node has an index ''i'', its children are found at indices $2i + 1$ (for the left child) and $2i +2$ (for the right), while its parent (if any) is found at index ''$\left \lfloor \frac \right \rfloor$'' (assuming the root has index zero). Alternatively, with a 1-indexed array, the implementation is simplified with children found at $2i$ and $2i+1$, and parent found at $\lfloor i/2 \rfloor$. This method benefits from more compact storage and better
locality of reference In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of Algori ...
, particularly during a preorder traversal. However, it is expensive to grow and wastes space proportional to 2''h'' - ''n'' for a tree of depth ''h'' with ''n'' nodes. This method of storage is often used for
binary heap A binary heap is a heap (data structure), heap data structure that takes the form of a binary tree. Binary heaps are a common way of implementing priority queues. The binary heap was introduced by J. W. J. Williams in 1964, as a data structure for ... s.

# Encodings

## Succinct encodings

A succinct data structure is one which occupies close to minimum possible space, as established by information theoretical lower bounds. The number of different binary trees on $n$ nodes is $\mathrm_$, the $n$th
Catalan number In combinatorial mathematics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite set, finite Mathematical structure, structures. It is ... (assuming we view trees with identical ''structure'' as identical). For large $n$, this is about $4^$; thus we need at least about $\log_4^ = 2n$ bits to encode it. A succinct binary tree therefore would occupy $2n+o\left(n\right)$ bits. One simple representation which meets this bound is to visit the nodes of the tree in preorder, outputting "1" for an internal node and "0" for a leaf

If the tree contains data, we can simply simultaneously store it in a consecutive array in preorder. This function accomplishes this: function EncodeSuccinct(''node'' n, ''bitstring'' structure, ''array'' data) The string ''structure'' has only $2n + 1$ bits in the end, where $n$ is the number of (internal) nodes; we don't even have to store its length. To show that no information is lost, we can convert the output back to the original tree like this: function DecodeSuccinct(''bitstring'' structure, ''array'' data) More sophisticated succinct representations allow not only compact storage of trees but even useful operations on those trees directly while they're still in their succinct form.

## Encoding general trees as binary trees

There is a one-to-one mapping between general ordered trees and binary trees, which in particular is used by
Lisp Lisp (historically LISP) is a family of programming language A programming language is a formal language In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbo ...
to represent general ordered trees as binary trees. To convert a general ordered tree to a binary tree, we only need to represent the general tree in left-child right-sibling way. The result of this representation will automatically be a binary tree if viewed from a different perspective. Each node ''N'' in the ordered tree corresponds to a node ''N' '' in the binary tree; the ''left'' child of ''N' '' is the node corresponding to the first child of ''N'', and the ''right'' child of ''N' '' is the node corresponding to ''N'' 's next sibling --- that is, the next node in order among the children of the parent of ''N''. This binary tree representation of a general order tree is sometimes also referred to as a
left-child right-sibling binary tree Every multi-way or k-ary tree In graph theory, an ''m''-ary tree (also known as ''k''-ary or ''k''-way tree) is a rooted tree (graph theory), tree in which each node has no more than ''m'' children. A binary tree is the special case where '' ...
(also known as LCRS tree, doubly chained tree, filial-heir chain). One way of thinking about this is that each node's children are in a
linked list In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of , ... , chained together with their ''right'' fields, and the node only has a pointer to the beginning or head of this list, through its ''left'' field. For example, in the tree on the left, A has the 6 children . It can be converted into the binary tree on the right.
The binary tree can be thought of as the original tree tilted sideways, with the black left edges representing ''first child'' and the blue right edges representing ''next sibling''. The leaves of the tree on the left would be written in Lisp as: :(((N O) I J) C D ((P) (Q)) F (M)) which would be implemented in memory as the binary tree on the right, without any letters on those nodes that have a left child.

# Common operations There are a variety of different operations that can be performed on binary trees. Some are mutator operations, while others simply return useful information about the tree.

## Insertion

Nodes can be inserted into binary trees in between two other nodes or added after a
leaf node In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of Algo ...
. In binary trees, a node that is inserted is specified as to whose child it will be.

### Leaf nodes

To add a new node after leaf node A, A assigns the new node as one of its children and the new node assigns node A as its parent.

### Internal nodes Insertion on
internal node In computer science, a tree is a widely used abstract data type that simulates a hierarchical tree structure, with a root value and subtrees of children with a #Terminology, parent node, represented as a set of linked Node (computer science), nod ...
s is slightly more complex than on leaf nodes. Say that the internal node is node A and that node B is the child of A. (If the insertion is to insert a right child, then B is the right child of A, and similarly with a left child insertion.) A assigns its child to the new node and the new node assigns its parent to A. Then the new node assigns its child to B and B assigns its parent as the new node.

## Deletion

Deletion is the process whereby a node is removed from the tree. Only certain nodes in a binary tree can be removed unambiguously.

### Node with zero or one children Suppose that the node to delete is node A. If A has no children, deletion is accomplished by setting the child of A's parent to null. If A has one child, set the parent of A's child to A's parent and set the child of A's parent to A's child.

### Node with two children

In a binary tree, a node with two children cannot be deleted unambiguously. However, in certain binary trees (including
binary search tree In computer science, a binary search tree (BST), also called an ordered or sorted binary tree, is a Rooted tree, rooted binary tree whose internal nodes each store a key greater than all the keys in the node's left subtree and less than those in ... s) these nodes ''can'' be deleted, though with a rearrangement of the tree structure.

## Traversal

Pre-order, in-order, and post-order traversal visit each node in a tree by recursively visiting each node in the left and right subtrees of the root.

### Depth-first order

In depth-first order, we always attempt to visit the node farthest from the root node that we can, but with the caveat that it must be a child of a node we have already visited. Unlike a depth-first search on graphs, there is no need to remember all the nodes we have visited, because a tree cannot contain cycles. Pre-order is a special case of this. See
depth-first search Depth-first search (DFS) is an algorithm In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are used as spe ...

Contrasting with depth-first order is breadth-first order, which always attempts to visit the node closest to the root that it has not already visited. See
breadth-first search Breadth-first search (BFS) is an algorithm for traversing or searching Tree (data structure), tree or graph (data structure), graph data structures. It starts at the tree (data structure)#Terminology, tree root (or some arbitrary node of a g ...
for more information. Also called a ''level-order traversal''. In a complete binary tree, a node's breadth-index (''i'' − (2''d'' − 1)) can be used as traversal instructions from the root. Reading bitwise from left to right, starting at bit ''d'' − 1, where ''d'' is the node's distance from the root (''d'' = ⌊log2(''i''+1)⌋) and the node in question is not the root itself (''d'' > 0). When the breadth-index is masked at bit ''d'' − 1, the bit values and mean to step either left or right, respectively. The process continues by successively checking the next bit to the right until there are no more. The rightmost bit indicates the final traversal from the desired node's parent to the node itself. There is a time-space trade-off between iterating a complete binary tree this way versus each node having pointer/s to its sibling/s.

# References

## Bibliography

*
Donald Knuth Donald Ervin Knuth ( ; born January 10, 1938) is an American computer scientist A computer scientist is a person A person (plural people or persons) is a being that has certain capacities or attributes such as reason, morality, consciousnes ...
. ''
The Art of Computer Programming ''The Art of Computer Programming'' (''TAOCP'') is a comprehensive monograph written by computer scientist Donald Knuth that covers many kinds of Computer programming, programming algorithms and analysis of algorithms, their analysis. Knuth began ...
vol 1. Fundamental Algorithms'', Third Edition. Addison-Wesley, 1997. . Section 2.3, especially subsections 2.3.1–2.3.2 (pp. 318–348).

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Binary Tree JavaScript Implementation with source code
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