CMPT 225

1
CMPT 225

• Binary Search Trees

2
Objectives

• Understand tree terminology
• Understand and implement tree traversals
• Define the binary search tree property
• Implement binary search trees

3
Trees

• A set of nodes with a single starting point
• called the root
• Each node is connected by an edge to some other
  node
• A tree is a connected graph
• There is a path to every node in the tree
• A tree has one less edge than the number of nodes

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Is it a Tree?
NO! All the nodes are not connected
yes! (but not a binary tree)
yes!
NO! There is an extra edge (5 nodes and 5 edges)
yes! (its actually the same graph as the blue
one)
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Tree Relationships

• If there is an edge between two nodes u and v,
  and u is above v in the tree, then v is said to
  be a child of u, and u the parent of v
• A is the parent of B, C and D
• This relationship can be generalized
• E and F are descendants of A
• D and A are ancestors of G
• B, C and D are siblings

6
More Tree Terminology

• A leaf is a node with no children
• A path is a sequence of nodes v1 vn
• where vi is a parent of vi1 (1 ? i ? n)
• A subtree is any node in the tree along with all
  of its descendants
• A binary tree is a tree with at most two children
  per node
• The children are referred to as left and right
• We can also refer to left and right subtrees

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Tree Terminology Example
A
path from A to D to G
B
D
subtree rooted at B
leaves C,E,F,G
G
E
F
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Binary Tree
A
B
C
right child of A
left subtree of A
F
G
D
E
right subtree of C
H
I
J
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Measuring Trees

• The height of a node v is the length of the
  longest path from v to a leaf
• The height of the tree is the height of the root,
  which is the length of the longest path from the
  root to a leaf
• The depth of a node v is the length of the path
  from the root to v
• This is also referred to as the level of a node
• Note that there are slightly different
  formulations of the height of a tree
• Where the height of a tree is said to be the
  number of different levels of nodes in the tree
  (counting the root)

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Height of a Binary Tree
A
height of the tree is 3
B
C
B
height of node B is 2
level 1
F
G
D
E
E
level 2
depth of node E is 2
H
I
J
level 3
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Binary Tree Descriptions

• A binary tree is perfect if no node has only one
  child and if all the leaves have the same depth
• A perfect binary tree of height h has (2h1 1)
  nodes, of which 2h are leaves
• A complete binary tree is one where
• The leaves are on at most two different levels,
• The second to bottom level is filled in and
• The leaves on the bottom level are as far to the
  left as possible.
• A balanced binary tree is one where
• No leaf is more than a certain amount farther
  from the root than any other leaf, this is
  sometimes stated more specifically as
• The height of any nodes right subtree is at most
  one different from the height of its left subtree
• A balanced trees height is sometime specified in
  terms of its relation to the number of nodes in
  the tree (see red-black trees)

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Perfect and Complete Binary Trees
Complete binary tree
Perfect binary tree
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Balanced Binary Trees
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Unbalanced Binary Trees
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Binary Tree Traversals

• A traversal algorithm for a binary tree visits
  each node in the tree
• and, typically, does something while visiting
  each node!
• Traversal algorithms are naturally recursive
• There are three traversal methods
• Inorder
• Preorder
• Postorder

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InOrder Traversal Algorithm

• // InOrder traversal algorithm
• inOrder(Node n)
• if (n ! null)
• inOrder(n.leftChild)
• visit(n)
• inOrder(n.rightChild)

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PreOrder Traversal
visit(n)
1
preOrder(n.leftChild)
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preOrder(n.rightChild)
visit preOrder(l) preOrder(r)
visit preOrder(l) preOrder(r)
13
27
2
6
3
9
39
16
20
5
7
8
visit preOrder(l) preOrder(r)
visit preOrder(l) preOrder(r)
visit preOrder(l) preOrder(r)
visit preOrder(l) preOrder(r)
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4
visit preOrder(l) preOrder(r)
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PostOrder Traversal
postOrder(n.leftChild)
8
postOrder(n.rightChild)
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visit(n)
postOrder(l) postOrder(r) visit
postOrder(l) postOrder(r) visit
13
27
4
7
9
39
16
20
2
3
5
6
postOrder(l) postOrder(r) visit
postOrder(l) postOrder(r) visit
postOrder(l) postOrder(r) visit
postOrder(l) postOrder(r) visit
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1
postOrder(l) postOrder(r) visit
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Binary Tree Implementation

• The binary tree ADT can be implemented using a
  number of data structures
• Reference structures (similar to linked lists)
• Arrays
• We will look at three implementations
• Binary search trees (references)
• Red black trees
• Heap (arrays)

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Problem Accessing Sorted Data

• Consider maintaining data in some order
• The data is to be frequently searched on the sort
  key e.g. a dictionary
• Possible solutions might be
• A sorted array
• Access in O(logn) using binary search
• Insertion and deletion in linear time
• An ordered linked list
• Access, insertion and deletion in linear time

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Dictionary Operations

• The data structure should be able to perform all
  these operations efficiently
• Create an empty dictionary
• Insert
• Delete
• Look up
• The insert, delete and look up operations should
  be performed in O(logn) time

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Binary Search Trees (BSTs)

• A binary search tree is a binary tree with a
  special property
• For all nodes v in the tree
• All the nodes in the left subtree of v have
  labels less than the label of v and
• All the nodes in the right subtree of v have
  labels greater than or equal to the label of v
• Binary search trees are fully ordered

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BST Example
17
13
27
9
39
16
20
11
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BST InOrder Traversal
inOrder(n.leftChild)
5
visit(n)
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inOrder(n.rightChild)
inOrder(l) visit inOrder(r)
inOrder(l) visit inOrder(r)
13
27
3
7
9
39
16
20
1
4
6
8
inOrder(l) visit inOrder(r)
inOrder(l) visit inOrder(r)
inOrder(l) visit inOrder(r)
inOrder(l) visit inOrder(r)
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2
inOrder(l) visit inOrder(r)
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BST Implementation

• Binary search trees can be implemented using a
  reference structure
• Tree nodes contain data and two references to
  nodes

Node leftChild
Node rightChild
Object data
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BST Search

• To find a value in a BST search from the root
  node
• If the target is less than the value in the node
  search its left subtree
• If the target is greater than the value in the
  node search its right subtree
• Otherwise return true, or return data, etc.
• How many comparisons?
• One for each node on the path
• Worst case height of the tree 1

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BST Search Example
click on a node to show its value
17
13
27
9
39
16
20
11
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BST Insertion

• The BST property must hold after insertion
• Therefore the new node must be inserted in the
  correct position
• This position is found by performing a search
• If the search ends at the (null) left child of a
  node make its left child refer to the new node
• If the search ends at the right child of a node
  make its right child refer to the new node
• The cost is about the same as the cost for the
  search algorithm, O(height)

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BST Insertion Example
insert 43 create new node find position insert
new node
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BST Deletion

• After deleting a node the BST property must still
  hold
• Deletion is not as straightforward as search or
  insertion
• So much so that sometimes it is not even
  implemented!
• There are a number of different cases that have
  to be considered

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BST Deletion Cases

• The node to be deleted has no children
• Remove it (assign null to its parents reference)
• The node to be deleted has one child
• Replace the node with its subtree
• The node to be deleted has two children
• Replace the node with its predecessor, the right
  most node of its left subtree (or with its
  successor, the left most node of its right
  subtree)
• If that node has a child (and it can have at most
  one child) attach that to the nodes parent

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BST Deletion target is a leaf
delete 30
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BST Deletion target has one child
delete 79 replace with subtree
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BST Deletion target has one child
delete 79 after deletion
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BST Deletion target has 2 children
delete 32
find successor and detach
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BST Deletion target has 2 children
delete 32
find successor
attach target nodes children to successor
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BST Deletion target has 2 children
delete 32
find successor
attach target nodes children to successor
make successor child of targets parent
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BST Deletion target has 2 children
delete 32
note successor had no subtree
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BST Deletion target has 2 children
Note predecessor used instead of successor to
show its location - an implementation would have
to pick one or the other
delete 63
find predecessor - note it has a subtree
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BST Deletion target has 2 children
delete 63
find predecessor
attach predecessors subtree to its parent
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BST Deletion target has 2 children
delete 63
find predecessor
attach subtree
attach targets children to predecessor
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BST Deletion target has 2 children
delete 63
find predecessor
attach subtree
attach children
attach predecssor to targets parent
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BST Deletion target has 2 children
delete 63
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BST Efficiency

• The efficiency of BST operations depends on the
  height of the tree
• All three operations (search, insert and delete)
  are O(height)
• If the tree is complete the height is ?log(n)?
• What if it isnt complete?

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Height of a BST

• Insert 7
• Insert 4
• Insert 1
• Insert 9
• Insert 5
• Its a complete tree!

height ?log(5)? 2
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Height of a BST

• Insert 9
• Insert 1
• Insert 7
• Insert 4
• Insert 5
• Its a linked list!

height n 1 4 O(n)
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Balanced BSTs

• It would be ideal if a BST was always close to
  complete
• i.e. balanced
• To guarantee a balance BST we would have to make
  the insertion and deletion algorithms more
  complex
• e.g. red black trees.