TREE ADT

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TREE ADT

• Department of Computer Science
• Xavier University

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Trees

• collection of elements called nodes, one of which
  is distinguished as a root, along with a relation
  (parenthood)
• imposes a hierarchical structure on a collection
  of items
• e.g. organizational charts

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Trees
C\
CS
MATH
CS20
CS22
MA5
MA6
Directory Structure represented by a Tree
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Trees

• a tree T is a set of nodes storing elements in a
  parent-child relationship with the following
  properties
• T has a distinguished node r, called the root of
  T, that has no parent
• each node v of T distinct from r has a parent
  node u
• for every node v of T, the root r is an ancestor
  of v

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Trees

• a single node by itself is a tree (this node is
  also the root)

A
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Trees

• an ancestor of a node is either the node itself
  or an ancestor of the parent of the node
• a node v is a descendent of a node u if u is an
  ancestor of v

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Trees
A is an ancestor of E E is a descendant of A
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Trees

• if node u is the parent of node v, then we say
  that v is a child of u
• two nodes that are children of the same parent
  are siblings

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Trees
B and C are children of A B and C are siblings
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Trees

• a node is external if it has no children
• a node is internal if it has one or more children
• external nodes are also called leaves

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Trees
A and B are internal nodes D, E, and C external
nodes
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Trees

• The subtree of T rooted at node v is the tree
  consisting of all the descendants of v in T
  (including v itself)

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Trees
T2 is a subtree of T1 rooted at B
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Trees

• a tree is ordered if there is a linear ordering
  defined for the children of each node in such a
  way that we can identify children of a node as
  being the first, second, third, and soon

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Trees
Book
Part 1
Part 2
Sec 1.1
Sec 1.2
Sec 2.1
Sec 2.2
Book Structure represented by an Ordered Tree
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Trees

• Binary Tree
• an ordered tree with each internal nodes having
  at most 2 children.
• Proper BT 0 or 2 children
• recursive definition BT is either
• an external node (leaf) or
• an internal node (the root) and two binary trees
  (left subtree and right subtree)

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Trees

• Binary Trees Properties
• ( external nodes ) ( internal nodes) 1
• ( nodes at level i) ? 2i
• ( external nodes) ? 2(height)
• (height) log2 ( external nodes)
• (height) log2 ( nodes) 1
• (height) ? ( internal nodes) (( nodes) - 1)/2

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Trees

• Binary Trees

Level
A
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Trees
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Trees

• the depth of v is the number of ancestors of v,
  excluding v itself
• the depth of a root is 0

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Trees
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Trees

• the height of a node v is
• if v is an external node, the height of v is 0
• otherwise, the height of v is one plus the
  maximum height of a child of v
• the height of an entire tree is height of the
  root
• height maximum depth of a node

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Trees
The height of B is 1 The height of the entire
tree is 2
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Trees

• Tree ADT Methods
• generic container methods
• size(), isEmpty(), elements()
• positional container methods
• positions(), swapElements(p,q),
  replaceElement(p,e)

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Trees

• Tree ADT Methods
• query methods
• isRoot(p), isInternal(p), isExternal(p)
• accessor methods
• root(), parent(p), children(p)

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Trees

• Binary Tree ADT Methods
• accessor methods
• leftChild(p), rightChild(p), sibling(p)
• update methods
• expandExternal(p), removeAboveExternal(p)

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Trees

• Tree Traversals
• a systematic way of accessing or visiting all the
  nodes of T
• basic traversal schemes
• Preorder Traversal
• Postorder Traversal
• Inorder Traversal

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Trees

• Preorder Traversal
• the root of T is visited first and then the
  subtrees rooted at its children are traversed
  recursively

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Trees
A B D E C
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Trees

• Algorithm preorder(T, v)
• perform the visit action for node v
• for each child w of v do
• recursively traverse the subtree rooted at w
  by calling preorder(T,w)

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Trees

• Preorder Traversal
• if the tree is ordered, then the subtrees are
  traversed according to the order of the children

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Trees

• Postorder Traversal
• opposite of the preorder traversal because it
  recursively traverses the subtrees rooted at the
  children of the root first, and then visits the
  root

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Trees
D E B C A
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Trees

• Algorithm postorder(T, v)
• for each child w of v do
• recursively traverse the subtree rooted at w
  by calling preorder(T,w)
• perform the visit action for node v

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Trees

• Inorder Traversal
• visit a node between the recursive traversals of
  its left and right subtrees

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Trees
D B E A C
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Trees

• Algorithm inorder(T, v)
• if v is an internal node then
• inorder(T,T.leftChild(v))
• perform the visit action for node v
• if v is an internal node then
• inorder(T,T.rightChild(v))

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Trees

• Sequence-Based Structure for B.T.
• based on a certain way of numbering the nodes of
  T
• node v of T is associated with the element of
  sequence S at rank p(v)
• typically, implemented using an array

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Trees
1
2
3
4
5
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7
8
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Binary Tree General Scheme
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Trees

• Sequence-Based Structure for B.T.
• for every node v of T, let p(v) be the integer
  defined
• if v is the root of T, then p(v)1
• if v is the left child of node u, then p(v)2p(u)
• if v is the right child of node u, then
  p(v)2p(u)1

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Trees

• Sequence-Based Structure for B.T.
• function p is known as a level numbering of the
  nodes in a binary tree

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Trees

• Linked Structure for Binary Trees
• each node v of T is represented by an object with
  references to the element stored at v and to the
  objects associated with the children and parent
  of v

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Trees
parent
Left Child
Right Child
element
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Trees
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Root
Size
Part 1
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Chap. 2
Chap. 1
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Section 1.2
Section 1.1
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Trees
parent
element
Children Container
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Trees
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4
Root
Size
Chap. 1
Sec. 1.1
Sec. 1.2
Sec. 1.3
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