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EE441 Data Structures Chapter IX Recursion, Trees

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AB-CD/EF-* Note: Postorder traversal, with the following ... Verify as an exercise! BA'2007. EE-441 Data Structures. 17. Tree Traversal Algorithms ... – PowerPoint PPT presentation

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Title: EE441 Data Structures Chapter IX Recursion, Trees


1
EE441 Data StructuresChapter IXRecursion, Trees
  • Özgür B. Akan
  • Department of Electrical Electronics
    Engineering
  • Middle East Technical University
  • akan_at_eee.metu.edu.tr
  • www.eee.metu.edu.tr/akan

2
Recursive Functions
  • Some calculations have recursive character
  • For example, N!N(N-1)!
  • To be able to implement such algorithms, we write
    functions that call themselves, i.e., recursive
    functions
  • Recursion
  • Method/Function calls itself
  • Each call is closer to Base Case
  • Base Case Termination Condition
  • Each call Loop iteration
  • Each call subject to stacking

3
Recursion-based Processing
  • ... doIt(..............)
  • if (there is data) // EOF processing
  • do something
  • ...
  • doIt(..........)

The recursive call... doIt( ) calls itself...
4
Recursion-based Processing
  • ... doIt(... int count ...)
  • if(count lt terminalValue)
  • do something
  • ...
  • doIt(... count 1 ...)

The recursive call... doIt( ) calls itself...
The value of count approaches the terminalValue
each call
5
Filling an Array - Recursion
  • // int howmany is global and initialized to 0
  • ... void getinput(... array ) ...
  • int num
  • fin gtgt num
  • if (!fin.eof( )...)
  • arrayhowmany num
  • howmany
  • getinput(array)

6
Finding the Sum - Recursion
  • // sum is declared globally
  • ... void findSum(int array , int x, int
    howmany)...
  • if(x lt howmany)
  • sum arrayx
  • findSum(array, x1, howmany)

The value of x approaches howmany each call
7
Trees
  • A tree is a set of nodes which is either null or
    with one node designated as the root and the
    remaining nodes partitioned into smaller trees,
    called sub-trees.
  • e.g.,
  • T1 (NULL Tree)
  • T2a a is the root, the rest is T1
  • T3a, b,c,d,e,f,g,h,i

8
Trees
  • Graphical representation of
  • T1 (NULL Tree)
  • T2a a is the root, the rest is T1
  • T3a, b,c,d,e,f,g,h,i

9
Trees
  • The level of a node is the length of the path
    from the root to that node
  • The depth of a tree is the maximum level of any
    node in the tree
  • The degree of a node is the number of partitions
    in the subtree which has that node as the root
  • Nodes with degree0 are called leaves

10
Binary Trees
  • A tree in which the maximum degree of any node is
    2.
  • A binary tree may contain up to 2n nodes at level
    n.
  • A complete binary tree of depth N has 2k nodes at
    levels k0,,N-1, and all leaf nodes at level N
    occupy leftmost positions.
  • If level N has 2N nodes as well, then the
    complete binary tree is a full tree.
  • If all nodes have degree1, the tree is a
    degenerate tree (or simply linked list)

level 0
level 1
level 2
level 3
level 4
11
Binary Trees - Examples
  • A degenerate tree of depth 5 has 6 nodes
  • A full tree of depth 3 has
  • 124815 nodes
  • A full tree of depth N has
  • 2N1-1 nodes
  • A complete tree of depth N has
  • 2N-1ltm 2N1-1 nodes or
  • 2Nm 2N1-1 nodes.
  • Exercise What is the depth of a complete tree
    with m nodes?

Show these as exercise
12
Data Structures and Representations of Trees
Left data right
T
?
?
?
?
?
  • template ltclass Tgt
  • class TreeNode
  • private
  • TreeNodeltTgt left
  • TreeNodeltTgt right
  • public
  • T data
  • //constructor
  • TreeNode(const T item, TreeNodeltTgt
    lptrNULL, TreeNodeltTgt rptrNULL)
  • //access methods for the pointer fields
  • TreeNodeltTgt Left(void) const
  • TreeNodeltTgt Right(void) const

13
Data Structures and Representations of Trees
  • template ltclass Tgt //constructor
  • TreeNodeltTgt TreeNode(const T item, TreeNodeltTgt
    lptr,
  • TreeNodeltTgt rptr) data(item), left(lptr),
    right(rptr)
  • // a function dynamically allocate memory for a
    new object
  • template ltclass Tgt
  • TreenodeltTgt GetTreeNode(T item, TreeNodeltTgt
    lptrNULL, TreeNodeltTgt rptrNULL)
  • TreeNodeltTgt p
  • pnew TreeNodeltTgt (item, lptr, rptr)
  • if (pNULL) // if "new" was unsuccessful
  • cerrltltMemory allocation failure"ltltendl
  • exit(1)
  • return p
  • // a function to deallocate memory template
    ltclass Tgt
  • void FreeTreeNode(TreeNode ltTgt p)

14
Data Structures and Representations of Trees
  • Example
  • TreeNodeltchargt t
  • tGetTreeNode('a', GetTreeNode('b',NULL,
    GetTreeNode('c')),
  • GetTreeNode('d', GetTreeNode('e')))

Result
15
Tree Traversal Algorithms
  • DEPTH-FIRST traversal algorithms
  • Inorder
  • Traverse left subtree
  • Visit node (i.e. process node)
  • Traverse right subtree
  • Preorder
  • Visit node
  • Traverse Left
  • Traverse right
  • Post-order
  • Traverse left
  • Traverse right
  • Visit node

16
Tree Traversal Algorithms
  • e.g. An arithmetic expression tree stores
    operands in leaves, operators in non-leaf nodes

Inorder traversal (LNR) (A-B)((C/D)(E-F)) A-BC
/DE-F (parentheses assumed) Preorder traversal
(NLR) -AB/CD-EF Postorder traversal
(LRN) AB-CD/EF-
If operand PUSH
Note Postorder traversal, with the following
implementation of visit
If operator POP two operands calculate
PUSH result back
Verify as an exercise!
corresponds to arithmetic expression evaluation!
17
Tree Traversal Algorithms
  • e.g. Counting leaves in a tree

template ltclass Tgt void CountLeaf(TreeNodeltTgt t,
int count) if (t !NULL) // using
postorder traversal CountLeaf(t-gtLeft(),count)
CountLeaf(t-gtRight(),count) // visiting a
node means incrementing if leaf if
(t-gtLeft()NULL t-gtRight()NULL) count

18
Tree Traversal Algorithms
  • e.g. Computing depth of a tree

template ltclass Tgt int Depth(TreeNodeltTgt t)
int depthLeft, depthRight, depthval if
(tNULL) depthval-1 // if empty, depth-1
else depthLeftDepth(t-gtleft()) depthRig
htDepth(t-gtRight()) depthval1(depthLeftgtdepth
Right ? depthLeftdepthRight)) return
depthval
Conditional expression syntax CONDITION?T-EXPF-E
XP
if true T-EXP, else F-EXP returned!
19
Breadth-First Traversal
  • The preorder, postorder, and inorder traversals
    are depth-first.
  • A Breadth-first traversal algorithm
  • e.g. Level-traverse traversal sequence a,
    b, c, d, e, f
  • Algorithm Level-Traverse
  • Insert root node in queue
  • While queue is not empty
  • Remove front node from queue and visit it
  • Insert Left child
  • Insert right child

20
Breadth-First Traversal
  • template ltclass tgt
  • void LevelScan(TreeNodeltTgt t)
  • QueueltTreeNodeltTgtgt Q // queue of pointers
  • TreeNodeltTgt p
  • // insert root
  • Q.Qinsert(t)
  • while(!Q.Empty())
  • pQ.Qdelete()
  • visit(p-gtdata)
  • if (p-gtLeft()!NULL) Q.Qinsert(p-gtLeft())
  • if (p-gtRight()!NULL) Q.Qinsert(p-gtRight())

21
Binary Search Trees
  • A BST is a BT in which data values in the left
    subtree of every node are less than the data
    value in the node and those in the right subtree
    are greater.
  • e.g.
  • Rather than using the full data field, a key
    field whose value must be unique in every node,
    is used for constructing the BST.
  • NOTE In a BST, searching and inserting are
    simple, but deleting may be problematic. -what
    to do when a root is deleted?

NOTE Inorder traversal generates ascending
sequence
22
Multiway Search Trees
  • Definition An m_way search tree is a tree in
    which all nodes are of degreeltm. (It may be
    empty).
  • A non empty m_way search tree has the following
    properties
  • a) It has nodes of type
  • b) key1 lt key2 lt...lt key(m-1) in other words,
  • key ilt key(i1), 1ltiltm-1
  • c) All Key values in subtree Ti are greater than
    Key i-1 and less than Key i1
  • Sometimes, we have an additional entry at the
    leftmost field of every node, indicating the
    number of nonempty key values in that node.

23
Multiway Search Trees
  • e.g. 3_way search tree
  • Nodes will be of type

24
Balanced Trees (B_Trees)
  • A B_tree of order m is an m_way search tree
    (possibly empty) satisfying the following
    properties (if it is not empty)
  • a) All nodes other than the root node and leaf
    nodes have at least m/2 children,
  • b) The tree is balanced. (If we modify all link
    fields of leaf nodes to point to special nodes
    called failure nodes, these are at the same
    level)
  • e.g. B-Tree of order 3

25
Inserting Key to B-Trees
  • Insert a new key value x, into a B-tree
  • Always insert at the leaf nodes.
  • The resulting tree must also be a B-tree. (It
    must be balanced.)
  • If space is not available at the node into which
    insertion will be made
  • Create a hypothetical node of order n1
  • Split into two nodes, insert extracted center key
    into parent
  • Repeatedly apply this until no further splits
    necessary

26
Inserting Key to B-Trees
  • Example
  • 1) Insert 38 to the B-tree of the above example
  • First of all, we do a search for 38 in the given
    B-tree.
  • We hit the failure node marked with "" . The
    parent of that failure node has only one key
    value so it has space for another one.
  • Insert 38 there , add a new failure node, which
    is marked as "" in the following figure, and
    return.

27
Inserting Key to B-Trees
  • 2) Now, insert 55 to this B-tree.
  • We do the search and hit the failure node "".
    However, it's parent node does not have any space
    for a key value. Now, assume we create a new node
    instead of

So, we end up
/
28
Inserting Key to B-Trees
  • 3) Now, let us insert 37 to this B_tree
  • We search for 37, and hit a failure node between
    35 and 38. So, we have to create

29
Inserting Key to B-Trees
  • So, we end up with

40
If we can not insert to the root node, we split
and create a new root node. For example try to
insert 34, 32, and 33. Thus, the height of the
B-tree increases by 1 in such a case. Deletion
algorithm is much more complicated! It will not
be considered here.
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