binary tree representation and traversal
Download as PPTX, PDF0 likes77 views
The document provides a comprehensive overview of binary trees, including their structure, terminology, and traversal methods. It explains the representations of binary trees, including array and linked representations, and details three standard traversal techniques: preorder, inorder, and postorder, complete with algorithms for each method. Each traversal method is illustrated with iterative steps to guide the reader through the process of visiting nodes in a binary tree.
![REPRESENTION OF BINARY TREES
ARRAY REPRESENTATION
(a)The Root R Of T is stored in
TREE[1]
(b)If a node N occupies TREE[K]
then,
• Left child is stored in TREE[2*k]
• Right child is stored in
TREE[2*k+1]
(c)NULL is used to indicate an
empty subtree. TREE[1]=NULL](https://image.slidesharecdn.com/dstraversalrepresent-2-200902151405/85/binary-tree-representation-and-traversal-4-320.jpg)
![• INFO[K] contains the data at node N
• LEFT[K]=Left child of node N
• RIGHT[K]=Right child of node N
(a)The value ROOT=14 indicates that
harris is the root of the tree.
(b)LEFT[14]=9 indicates that cohen is
the left child of harris,
RIGHT[14]=7 indicates that lewis is the
right child of harris
(c)Repeating step (b) untill reach the
NULL
LINKED REPRESENTATION](https://image.slidesharecdn.com/dstraversalrepresent-2-200902151405/85/binary-tree-representation-and-traversal-5-320.jpg)
![PREORDER TRAVERSAL ALGORITHM
PREORD(INFO, LEFT, RIGHT, ROOT)
• [Initially push NULL onto STACK, andIinitialize PTR. ]
Set TOP:=1, STACK[1]:=NULL and PTR:=ROOT.
• Repeat Steps 3 to 5 while PTR!= NULL:
• Apply PROCESS to INFO[PTR].
• [Right child?]
If RIGHT[PTR]!=NULL, then:[Push on STACK. ]
Set TOP:=TOP+1, and STACK[TOP]:=RIGHT[PTR].
[End of If structure]
• [Left child? ]
If LEFT[PTR]!=NULL, then:
Set PTR:=LEFT[PTR].
Else:[Pop from STACK. ]
Set PTR:=STACK[TOP] and TOP:=TOP-1.
[End of if structure. ]
[End of step 2 loop. ]
• Exit.](https://image.slidesharecdn.com/dstraversalrepresent-2-200902151405/85/binary-tree-representation-and-traversal-8-320.jpg)
![INORDER TRAVERSAL
ITeRATION
1. STACK:NULL set PTR=A
2. STACK:NULL,A, B, D, G, K
3. STACK:NULL,A, B Set PTR:=H
4. PTR:H, STACK:NULL, A, B, H, L
5. STACK:NULL,A, B Set PTR:=M
6. STACK:NULL,A,B,M
7. STACK:NULL,PTR=C
8. PTR=C, STACK:NULL ,C, E
9. [BT]E, C
K, G, D, L, H, M, B, A, E, C](https://image.slidesharecdn.com/dstraversalrepresent-2-200902151405/85/binary-tree-representation-and-traversal-9-320.jpg)
![INORDER TRAVERSAL ALGORITHM
INORD(INFO, LEFT, RIGHT, ROOT)
• [Push NULL onto STACK and initialize PTR. ]
Set TOP:=1, STACK[1]:=NULL and
PTR:=ROOT.
• Repeat while PTR!=NULL:[Pushes left most
path onto stack. ]
(a) Set TOP:=TOP+1 and STACK[TOP]:=PTR.
[Saves node. ]
(b) Set PTR:=LEFT[PTR]. [Updates PTR. ]
[End of loop. ]
• Set PTR:=STACK[TOP] and TOP:=TOP-1.
[ Pops node from STACK. ]
• Repeat Steps 5 to 7 while
PTR!=NULL:[Backtracking. ]
• Apply PROCESS to INFO[PTR].
• [Right child?] If RIGHT[PTR]!=NULL, then:
(a) Set PTR:=RIGHT[PTR].
(b) Go to step 2.
[ End of if structure. ]
• Set PTR:=STACK[TOP] and TOP:=TOP-1.[Pops node]
[End of step 4 loop. ]
• Exit.](https://image.slidesharecdn.com/dstraversalrepresent-2-200902151405/85/binary-tree-representation-and-traversal-10-320.jpg)
![POSTORDER TRAVERSAL ALGORITHM
• Set PTR:=SRACK[TOP] and TOP:=TOP-
1.
[Pops node from STACK. ]
• Repeat while PTR>0;
(a) Apply PROCESS to INFO[PTR].
(b) Set PTR:=STACK[TOP] and
TOP:=TOP-1.
[ Pops node from STACK. ]
[End of loop. ]
• If PTR<0, then :
(a) Set PTR:=-PTR.
(b) Go to step 2.
[ End of if structure. )
• Exit.
POSTORD(INFO, LEFT, RIGHT, ROOT)
• [Push NULL onto STACK and initialize PTR.
]
Set TOP:=1, STACK[1]:=NULL and
PTR:=ROOT.
• [Push left most path onto stack. ]
Repeat Steps 3 to 5 while PTR!= NULL:
• Set TOP:=TOP+1 and STACK[TOP]:=PTR.
[Pushes PTR on STACK. ]
• If RIGHT[PTR]!=NULL, then:[Push on
STACK. ]
Set TOP:=TOP+1 and
STACK[TOP]:=RIGHT[PTR].
[End of If structure]
• Set PTR:=LEFT[PTR]. [Updates pointer](https://image.slidesharecdn.com/dstraversalrepresent-2-200902151405/85/binary-tree-representation-and-traversal-12-320.jpg)
binary tree representation and traversal
- 1. BINARY TREE REPRESENTATION AND TRAVERSAL 19IFTE063_K.S.Jothika 19IFTE073_T.G.Preethi
- 2. TREE • A Tree is a finite set of one or more nodes • (i). Root (ii). Subtree BASIC TERMINOLOGY • Root=M • Subtree=G, W • Leaves=B, H, J, N, S • Terminal Nodes=B, H, N, S • Non Terminal Nodes=M, G, W, D, P • Level=4 • Height/Depth=ex(J):2 • Degree=G, W • Parent=G, W, D, P, M • Child=G, W, D, J, P, B, H, N, S
- 3. BINARY TREE •A Binary Tree is set of nodes which is either empty or consists of a root and to disjoint binary trees called LEFT SUBTREE and RIGHT SUBTREE. Example of binary tree: SKEWED BINARY TREE COMPLETE BINARY TREE
- 4. REPRESENTION OF BINARY TREES ARRAY REPRESENTATION (a)The Root R Of T is stored in TREE[1] (b)If a node N occupies TREE[K] then, • Left child is stored in TREE[2*k] • Right child is stored in TREE[2*k+1] (c)NULL is used to indicate an empty subtree. TREE[1]=NULL
- 5. • INFO[K] contains the data at node N • LEFT[K]=Left child of node N • RIGHT[K]=Right child of node N (a)The value ROOT=14 indicates that harris is the root of the tree. (b)LEFT[14]=9 indicates that cohen is the left child of harris, RIGHT[14]=7 indicates that lewis is the right child of harris (c)Repeating step (b) untill reach the NULL LINKED REPRESENTATION
- 6. TRAVERSING BINARY TREES Tree Traversal is the process of visiting each node in the tree exactly once There are three standard ways of traversing a binary tree T with Root R. • Preorder • Inorder • postorder
- 7. PREORDER TRAVERSAL ITeRATION 1. STACK:NULL set PTR=A 2. STACK:NULL, C Process B 3. STACK:NULL, C Process D 4. STACK: NULL, C, H Process G 5. STACK:NULL, C PTR=H 6. STACK:NULL, C, K No Left child 7. STACK:NULL, C Process K 8. STACK:NULL Process C 9. STACK:NULL, F Process E 10. STACK:NULL, popout F and Process F A, B, D, G, H, K, C, E, F
- 8. PREORDER TRAVERSAL ALGORITHM PREORD(INFO, LEFT, RIGHT, ROOT) • [Initially push NULL onto STACK, andIinitialize PTR. ] Set TOP:=1, STACK[1]:=NULL and PTR:=ROOT. • Repeat Steps 3 to 5 while PTR!= NULL: • Apply PROCESS to INFO[PTR]. • [Right child?] If RIGHT[PTR]!=NULL, then:[Push on STACK. ] Set TOP:=TOP+1, and STACK[TOP]:=RIGHT[PTR]. [End of If structure] • [Left child? ] If LEFT[PTR]!=NULL, then: Set PTR:=LEFT[PTR]. Else:[Pop from STACK. ] Set PTR:=STACK[TOP] and TOP:=TOP-1. [End of if structure. ] [End of step 2 loop. ] • Exit.
- 9. INORDER TRAVERSAL ITeRATION 1. STACK:NULL set PTR=A 2. STACK:NULL,A, B, D, G, K 3. STACK:NULL,A, B Set PTR:=H 4. PTR:H, STACK:NULL, A, B, H, L 5. STACK:NULL,A, B Set PTR:=M 6. STACK:NULL,A,B,M 7. STACK:NULL,PTR=C 8. PTR=C, STACK:NULL ,C, E 9. [BT]E, C K, G, D, L, H, M, B, A, E, C
- 10. INORDER TRAVERSAL ALGORITHM INORD(INFO, LEFT, RIGHT, ROOT) • [Push NULL onto STACK and initialize PTR. ] Set TOP:=1, STACK[1]:=NULL and PTR:=ROOT. • Repeat while PTR!=NULL:[Pushes left most path onto stack. ] (a) Set TOP:=TOP+1 and STACK[TOP]:=PTR. [Saves node. ] (b) Set PTR:=LEFT[PTR]. [Updates PTR. ] [End of loop. ] • Set PTR:=STACK[TOP] and TOP:=TOP-1. [ Pops node from STACK. ] • Repeat Steps 5 to 7 while PTR!=NULL:[Backtracking. ] • Apply PROCESS to INFO[PTR]. • [Right child?] If RIGHT[PTR]!=NULL, then: (a) Set PTR:=RIGHT[PTR]. (b) Go to step 2. [ End of if structure. ] • Set PTR:=STACK[TOP] and TOP:=TOP-1.[Pops node] [End of step 4 loop. ] • Exit.
- 11. POSTORDER TRAVERSAL ITeRATION 1. STACK:=NULL set PTR:=A 2. STACK:=NULL,A,-C, B,D,-H, G, K 3. POP K, G, -H STACK:NULL,A, -C, B, D PTR=-H Reset PTR:=H 4. PTR:H, STACK:NULL, A,-C, B, D, H, -M, L 5. Pop L, -M 6. PTR=M STACK:NULL,A,-C, B, D, H, M Pop M, H, D, B, -C PTR=C 7. PTR=C, STACK:NULL ,A, C, E Pop and Process E, C, AK, G, L, M, H, D, B, E, C, A
- 12. POSTORDER TRAVERSAL ALGORITHM • Set PTR:=SRACK[TOP] and TOP:=TOP- 1. [Pops node from STACK. ] • Repeat while PTR>0; (a) Apply PROCESS to INFO[PTR]. (b) Set PTR:=STACK[TOP] and TOP:=TOP-1. [ Pops node from STACK. ] [End of loop. ] • If PTR<0, then : (a) Set PTR:=-PTR. (b) Go to step 2. [ End of if structure. ) • Exit. POSTORD(INFO, LEFT, RIGHT, ROOT) • [Push NULL onto STACK and initialize PTR. ] Set TOP:=1, STACK[1]:=NULL and PTR:=ROOT. • [Push left most path onto stack. ] Repeat Steps 3 to 5 while PTR!= NULL: • Set TOP:=TOP+1 and STACK[TOP]:=PTR. [Pushes PTR on STACK. ] • If RIGHT[PTR]!=NULL, then:[Push on STACK. ] Set TOP:=TOP+1 and STACK[TOP]:=RIGHT[PTR]. [End of If structure] • Set PTR:=LEFT[PTR]. [Updates pointer