The document provides a comprehensive overview of binary trees, defining key concepts such as nodes, edges, and sub-trees, along with the structure of ordered and unordered trees. It explains tree traversals, including pre-order, post-order, and in-order methods, as well as insertion and deletion processes in both ordered and unordered contexts. Additionally, it touches on coding practices for tree structures, including methods for insertion, searching, and traversal outputs.

1. BINARY TREES – UNIT REVIEW PAGE 1 OF 2 COMPUTER SCIENCE
BINARY TREES – UNIT REVIEW TUESDAY, APRIL 25, 2017 COMPUTER SCIENCE
D E F I N I T I O N :
 Trees are data structures used to represent relationships, that are hierarchical in nature, meaning
that a ‘parent’ and ‘child’ relationship exists in the tree
 Nodes (or vertices) are shown with paths (or edges) between them:
 Each node in a tree has one parent, and one node (only one) has no parents, this node is called the
root of the tree
 Nodes that have no children are known as leaf nodes
 The paths between nodes are known as branches (or edges)
 The terminology for trees can also be extended so that we can say that A is an ancestor of D, E,
and F and D is a descendant of F, and the root (A) is an ancestor of all nodes in the tree
 In-Degree: means the number of branches that lead into a node, in this tree, the root has degree 0, all other nodes have degree 1
 Out-degree: the number of branches that lead out of a node, leaf nodes have out-degree 0, all other nodes >= 1
 A sub-tree is a portion of the larger tree; it is defined as a node together with all of its descendants
B I N A R Y T R E E :
 Binary trees are defined as a set of nodes T in which: T is empty, or T is partitioned into three disjoint
subsets: a single node r, the root, and two possibly empty sets that are binary trees, called the left sub-
tree TL and right sub-tree TR of the root r
 So simply, T is a binary tree if it
has no nodes, or is of the form:
 A full binary tree is when every
node up to a certain level is
present:
 In a binary tree, every node has 0,
1 or 2 children, and in a full binary
tree, every node has 0 or 2
children
 In a binary tree, each node can
have a left sub-tree and a right
sub-tree attached to it
 In ordered binary trees, every node is
arranged so that for every node, the node item is less
than (<) the all the node items in it’s left sub-tree and is
greater than (>=) all the node items in the right sub-tree
S E A R C H I N G A N D T R A V E R S I N G A T R E E :
 If a tree is unordered, then to search, you must traverse the whole tree node by node
 However, if a tree is ordered, you can use a binary search of the tree to increase efficiency, as well this can be done recursively
 To ‘traverse’ a tree means to visit every node (traverse algorithms are the same for ordered and unordered trees)
 The reasons for traversals could include: printing all node values, counting nodes, searching the nodes, etc.
 There are 3 orders to the traversals: (1) Pre-order (means you visit ‘before’ left, right), (2) Post-order (go left, right, then visit ‘after’),
(3) In-order (visit in-between left and right)
 These traversals can also be accomplished recursively
 Pre-order traversal: (1) Visit the current node (starting at the root), (2) Traverse the left sub-tree, (3) Traverse the right sub-tree
 Post-order traversal: (1) Traverse the left sub-tree, (2) Traverse the right sub-tree, (3) Visit the current node (starting at the root)
 In-order traversal: (1) Traverse the left sub-tree, (2) Visit the current node (starting at the root), (3) Traverse the right sub-tree
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BINARY TREES – UNIT REVIEW
Sub-tree

2. BINARY TREES – UNIT REVIEW PAGE 2 OF 2 COMPUTER SCIENCE
BINARY TREES – UNIT REVIEW TUESDAY, APRIL 25, 2017 COMPUTER SCIENCE
I N S E R T I N G A N D R E M O V I N G F R O M A T R E E :
 If a tree is unordered, then to search, you must traverse the whole
tree node by node
 With an unordered tree, you can insert a node anywhere you have a
null
 With an ordered binary tree, there is only one spot you can insert a new node
(which you search for) – for example, insert: 15, 8, 12, 19, 5, 23
 If you receive the numbers in a different order, you build a different shaped binary
search tree (BST)
 Removing a node from a tree can be more difficult, and three scenarios exist: (1)
The node has 0 children, (2) The node has 1 child, (3) The node has 2 children
 To remove a leaf (0 children) you just set the pointer to the leaf node to null
 To remove a node with 1 child, just redirect the pointer of the node pointing to the
node to be deleted to that node’s child
 To remove a node with 2 children, a shift of the tree needs to occur (this
complexity does not need to be examined here)
S A M P L E L I N K E D T R E E P R O P E R T I E S A N D M E T H O D S :
 The following list represents some methods and properties that could be coded into a Tree ADT to increase
its functionality:
− A TreeNode class different than the LinkedList Node class that has self-references to left and
right (not next and previous)
− A recursive insert(data) method in the TreeNode class that inserts a new node recursively to the
left or right based on if the data is “less than” (to the left), or “greater than or equal to” (to the right)
− Tree properties include a root TreeNode reference
− The search(data) → boolean method: determines if the data is in the tree or not
− The preOrder() method: traverses the list and ‘outputs’ the nodes in pre-order
− The postOrder() method: traverses the list and ‘outputs’ the nodes in post-order
− The inOrder() method: traverses the list and ‘outputs’ the nodes in-order
O T H E R T R E E F A C T S :
 Non-binary trees can contain more than two branches from
a node
 Another traversal you can do is a breadth-first traversal, in
which you visit nodes level by level, for example a breadth
first traversal of this tree (there is no easy recursive method
for doing this)