Introduction to Trees

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Introduction to Trees
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Tree example

• Consider this program structure diagram as itself
  a data structure.

main
readin
print
process
sort
lookup
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Terminology

• Node - an element of a tree
• Root node - the single node at level 0
• Child node - nodes at level 1 or lower
  (descending from a parent node). Note a node
  can have only one parent.

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Terminology (continued)

• Parent node - a node at one level associated with
  nodes at a lower level (children).
• Note nodes without children are called leaves.
• Every node of a tree is the root of a subtree.

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• As you can see by the diagram of a tree, its
  structure is highly recursive. Therefore, as you
  would expect, the procedures and functions that
  manipulate trees are also recursive in nature.

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The tree as an ADT

• Fundamental operationsCreate(Root)Empty(Root)Fi
  nd(Root,Target,TargetLoc)Add(Root,Node)Delete(Ro
  ot,Node)Traverse(Root)

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Binary Trees

• This (next slide) is an example of a binary tree
  (ie. one in which each node has no more than two
  children)

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Binary tree example
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(see previous tree)

• This tree also has the property that each node is
  greater than either of its children. Another
  name for it is a heap (not to be confused with
  the system heap).

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Question

• What is the hierarchical ordering principal for
  this tree?
• (see next slide)

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What is the ordering property?
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Answer

• Each node is greater than all of the nodes in its
  left subtree and less than all of the nodes in
  its right subtree.
• This is called its ordering property.

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Trees representing arithmetic expressions.

• For example (A-B)(C(E/F))

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Example
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• compare this tree to a tree for the expression
  (A-B)(CE/F)

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• Notice how small changes in the order of
  operations (like removing a set of parentheses)
  make major changes in the structure of the tree.

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Reverse Polish Again!?!

• Recall that expressions are easily evaluated
  using a stack if they are available in postfix
  form AB-CEF/
• Trees can be traversed in such a way as to
  produce the postfix form. eg. a postorder
  traversal yields the postfix form of the
  expression.
• More on this later.

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Implementing Binary Trees

• Two methods
• 1. Linear representation - array
• 2. Linked representation - pointers

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Linear representation (Depth3)

• 1. Allocate an array of size 2(d1)-1
• 2. Store root in location 1
• 3. For node in location n, store left child in
  location 2n, right child in location 2n1

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Tradeoffs

• 1. Fast access (given a node, its children and
  parent can be found very quickly)
• 2. Slow updates (inserts and deletions require
  physical reordering)
• 3. Wasted space (partially filled trees)

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Linked Representation

• Use records with pointer fields(A-B)(C(E/F))

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Using an array of records

• The array implementation of this ADT looks like
  this (next slide)
• Insertions and deletions involve only
  manipulation of pointers

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Array of structures
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Insertion (A-B)(P-C(E/F))
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Deletion (A-B)(P-E/F)

• The changes to the array of records look like
  this after both of the operations (insertion of
  -P and deletion of C) have been performed.

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Tradeoffs

• 1. Faster updates
• 2. Extra memory for pointers
• 3. Wasted space for nil pointers (can be
  corrected by threading)
• 4. Difficult to determine a node's parent
  (corrected by adding a parent pointer)

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Traversing Binary Trees

• A tree traversal visits each node of a tree
  exactly once

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Modes of traversal

• When visiting a node we have three choices
• 1. Process the data
• 2. Remember location of the current node and
  process nodes in the left subtree
• 3. Remember location of the current node and
  process nodes in the right subtreeThe choice
  made dictates the order of node processing for
  the entire tree

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Example tree (T)
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Steps for a Preorder traversal of a binary tree

• 1. Process root
• 2. Do a preorder traversal of the nodes in the
  root's left subtree(recursively)
• 3. Do a preorder traversal of the nodes in the
  root's right subtree (recursively)

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• Assume the we have a procedure called Process
  that simply writes out the data contained in any
  given node.
• Then, using process, a preorder traversal of tree
  T produces ABC/EF (the prefix form of
  (A-B)C(E/F))

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Steps for an Inorder traversal of a binary tree

• 1. Recursively visit left subtree
• 2. Process root
• 3. Recursively visit right sub-tree

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Steps for a Postorder traversal of a binary tree

• 1. Recursively visit left subtree
• 2. Recursively visit right subtree
• 3. Process Root

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• A postorder traversal of T produces AB- CEF/
  (the postfix form)
• These traversals can be done on any binary tree,
  not just expression trees.

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Implementing lists using binary trees
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Implementing the List ADT

• Vehicle Advantage Disadvantage
• Array Fast search Slow inserts
  O(log2N) and deletions
  O(N)
• Linked Fast inserts Slow search List and
  deletions O(N) O(1)
• Binary Fast inserts Tree must be
  Tree and deletions balanced O(log2N)
  The hierarchical relationship we employed
  is the ordering property.

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Example tree T1
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• Inorder traversal of this tree gives
  ADEFGHKMNPSTUVZSo An inorder traversal of a
  binary tree with the ordering property visits the
  nodes in ascending order.

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• Adding a node to a binary tree with the ordering
  property
• Follow the insertion rule "If less than, go
  left, otherwise go right."

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Example Add the letter R to the previous tree.

• 1. Compare R to M (the root). Since R M,
  add R to the right subtree
• 2. Compare R to T (the root of the subtree).
  Since R
• 3. Compare R to P. Since R P go right
• 4. Compare R to S. Since R
• 5. Since the left subtree is NULL add R at
  this pointClearly, the procedure is
  recursive.

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Adding R to the tree
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Efficiency of adding to a binary tree

• Question How many comparisons did we have to
  make?Answer 4 (ie. depth 1)

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Differing tree configurations

• Consider another tree T2 representing the same
  list

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Question

• How many comparisons do we have to make to add
  the letter R?Answer 9 (ie. depth 1)

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Question

• How do you minimize the depth of a binary
  tree?Answer Make sure it is full, or balanced.

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Definition

• A balanced binary tree is one in which every node
  above level depth-1 has exactly two children.

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Depth to size comparisons

• When a binary tree of N nodes is balanced, no
  branch will be longer than log(base 2)N1. Thus,
  the efficiency of the adding algorithm is
  O(log(base 2)N)
• Depth MaxN
• 0 1 1 3 2 7 3 15 . . .
  9 1023 10 1 million

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Suppose we add Q to T1
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Suppose we add Q to T1

• Now T1 is not balanced.
• There are methods for balancing trees.

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Recall the binary search of an array
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Binary similarities

• Each probe into the array is analogous to a
  recursive descent in the search of a binary tree.
  (both are O(log2N))

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Deleting from a tree
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For example, remove T from tree T1.

• The result becomes

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Deletion strategies

• Requirement The resulting tree must have the
  ordering property.
• One method is to find the largest item in T1's
  left sub-tree and replace T with it.

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Three special cases which might occur

• 1. Node to be deleted has a left child
• 2. Node to be deleted has a right child but
  no left child
• 3. Node to be deleted has no children

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Case 1

• Suppose P points to the node to be deleted. There
  are two important subcases
• 1.a P-Left has no right child
• 1.b P-Left has a right child

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Case 1a
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Case 1b
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Case 2
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Case 3
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List comparison

• Tradeoffs Linked List Binary Tree
• adding O(1) O(1) deleting searching O(N) O(
  log2N) balanced O(N) worst case
• user 1 ptr/node 2 ptrs/node memory (many
  NULL)
• system stack minimal recursion (remedy -
  threading)

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General Trees
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General Trees

• So far we have been studying binary trees,
  general trees however can have more than two
  children for each node.Question Does the
  former restriction to binary trees mean that we
  cannot apply what we have learned to general
  trees?Answer No! As long as we are willing
  to convert general trees to binary ones.

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Binary version of general tree
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Preorder Traversal

• (Root, Left, Right)
• J B D N E P Q K L M C X R T

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Processing order

• 1. Process root2. Recursively traverse the
  left-most subtree3. Recursively traverse the
  next subtree to the right4. Repeat step 3
  until there are no more subtrees

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Postorder Traversal

• (Left, Right, Root)
• Q P E N D L X R C T M K B J

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• Note that this order on the general tree can be
  described by1. Start at the leaves and work
  up2. Process a node after all nodes below it
  have been processed