TDDB57 DALG – Lecture 6: ADT Tree, Binary Search Trees
Lecture 6 – Overview
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J. Maluszynski, IDA, Linköpings Universitet, 2004.
[Lewis/Denenberg p.103]
[Lewis/Denenberg 4.4]
ADT Tree
Tree representations
TDDB57 DALG – Lecture 6: ADT Tree, Binary Search Trees
ADT Tree (1)
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J. Maluszynski, IDA, Linköpings Universitet, 2004.
Domain: tree nodes, maybe associated with additional information
Operations on a single tree node v:
[Lewis/Denenberg 4.3]
[Lewis/Denenberg pp.193-196, 198-199]
Tree traversal strategies
Binary search trees, implementation of Dictionary
Insertion and deletion take constant time.
RC
J. Maluszynski, IDA, Linköpings Universitet, 2004.
LC
RC
RC
RightChild
If required, a “backward” pointer to the parent node can be added.
Alternatively, the pointers to a node’s children can be stored in a linked list.
LC
LeftChild
Tree representations (1): using pointers
For binary trees:
2 pointers per node, LC and RC
[1..nchilds]
ADT Tree (2)
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Parent v returns parent of v, or Λ if v root
Children v returns set of children of v, or Λ if v leaf
FirstChild v returns first child of v, or Λ if v leaf
LeftChild v , RightChild v returns left / right child of v, or Λ if not existing
RightSibling v returns right sibling of v, or Λ if v is a rightmost child
LeftSibling v returns left sibling of v, or Λ if v is a leftmost child
IsLeaf v returns true iff v is a leaf
TDDB57 DALG – Lecture 6: ADT Tree, Binary Search Trees
Type Tnode denotes a pointer to a structure storing node information:
J. Maluszynski, IDA, Linköpings Universitet, 2004.
Operations on an entire tree T :
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record node record
nchilds: integer
child: table Tnode
info: infotype
TDDB57 DALG – Lecture 6: ADT Tree, Binary Search Trees
Size T returns number of nodes of T
Root T returns root node of T
IsRoot v T returns true iff v is root of T
Depth v T returns depth of v in T
Height v T returns height of v in T
Height T returns height of the root of T
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TDDB57 DALG – Lecture 6: ADT Tree, Binary Search Trees
Tree traversals (1)
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Regard a tree T as a building:
nodes as rooms, edges as doors, root as entry
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TDDB57 DALG – Lecture 6: ADT Tree, Binary Search Trees
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Tree traversals(3): Level-order traversal
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Traverse all the nodes of a tree:
by increasing depth
nodes of the same depth from left to right
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J. Maluszynski, IDA, Linköpings Universitet, 2004.
Level-order traversal implemented on Queue (see course book p.124)
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J. Maluszynski, IDA, Linköpings Universitet, 2004.
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How to explore an unknown (acyclic) labyrinth and get out again?
Proceed by always keeping a wall to the right!
Generic tree traversal routine:
procedure traverse ( node v )
explore subtree rooted at v
for all u Children v do
traverse u
exactly one call for each node in T ; recursive: needs Stack!!
Call traverse( Root(T ) ):
J. Maluszynski, IDA, Linköpings Universitet, 2004.
index positions
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TDDB57 DALG – Lecture 6: ADT Tree, Binary Search Trees
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J. Maluszynski, IDA, Linköpings Universitet, 2004.
Tree representations (2): array indexing
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LeftChild i : 2i 1
(none if 2i 1 n)
RightChild i : 2i 2
(none if 2i 1 n)
IsLeaf i : 2i 1 n
LeftSibling i : i 1
(none if i 0 or i odd)
RightSibling i : i 1
(none if i n 1 or i even)
Parent i : i 1 2 (none if i 0)
Depth i : log2 i 1
Height i : log2 n 1 i 1
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For a complete binary tree holds:
There is exactly one complete binary tree with n nodes.
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Implicit representation of edges: Numbering of nodes
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only for binary trees
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TDDB57 DALG – Lecture 6: ADT Tree, Binary Search Trees
Tree traversals (2)
procedure preorder visit ( node v )
output v
before any of the subtree nodes are output
for all u Children v do
preorder visit u
procedure postorder visit ( node v )
for all u Children v do
postorder visit u
output v
after all of the subtree nodes have been output
procedure inorder visit ( node v )
inorder visit LC v
output v
inorder visit RC v
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J. Maluszynski, IDA, Linköpings Universitet, 2004.
TDDB57 DALG – Lecture 6: ADT Tree, Binary Search Trees
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RC
LC
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RC
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RC
Insert v : add v as new leaf on look-up failure
worst-case: height T
1 comparisons
Delete v :
if v is a leaf, remove v
if v has one child u, replace v by u
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LookUp(22)
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1 comparisons
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[Lewis/Denenberg Alg. 6.8]
J. Maluszynski, IDA, Linköpings Universitet, 2004.
if v has two children, replace v by its inorder successor
(alternatively: by its inorder predecessor)
worst-case: height T
TDDB57 DALG – Lecture 6: ADT Tree, Binary Search Trees
LookUp in BST
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[Lewis/Denenberg Alg. 6.7]
J. Maluszynski, IDA, Linköpings Universitet, 2004.
Dictionary as a Binary Search Tree
TDDB57 DALG – Lecture 6: ADT Tree, Binary Search Trees
Implementing Dictionaries as Binary Search Trees
Not unique: same data may be represented by different BST
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J. Maluszynski, IDA, Linköpings Universitet, 2004.
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LookUp: comparison controlled traversal
worst-case: height T
1 comparisons
A binary search tree (BST) is a binary tree such that:
Information associated with a node includes a key,
linear ordering of nodes determined by keys.
LC
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The key of each node is:
greater than the keys of all left descendants, and
smaller than the keys of all right descendants.
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TDDB57 DALG – Lecture 6: ADT Tree, Binary Search Trees
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Different BST representing the same data
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TDDB57 DALG – Lecture 6: ADT Tree, Binary Search Trees
Insert in BST
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Delete(21)
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Insert(27)
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J. Maluszynski, IDA, Linköpings Universitet, 2004.
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J. Maluszynski, IDA, Linköpings Universitet, 2004.
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TDDB57 DALG – Lecture 6: ADT Tree, Binary Search Trees
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Delete in BST
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inorder
successor
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TDDB57 DALG – Lecture 6: ADT Tree, Binary Search Trees
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Delete(12)
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Successful LookUp: Worst and Average Case
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Worst Case BST
BST degenerated to a linear sequence
expected number of comparisons is n 1 2
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Balanced BST
the depths of leaves differ by at most 1
Apply the Binary Search Expected Time Theorem
O log2 n comparisons.
TDDB57 DALG – Lecture 6: ADT Tree, Binary Search Trees
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Delete in BST: an alternative approach
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inorder
predecessor
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J. Maluszynski, IDA, Linköpings Universitet, 2004.
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J. Maluszynski, IDA, Linköpings Universitet, 2004.