1. COSC2007 Data Structures II
April 21, 2017
COSC2007 Data Structures II
Chapter 11
Trees I
Trees

2. Topics
Terminology

3. Introduction & Terminology
Review
Position-oriented ADT:
Insert, delete, or ask questions about data items at specific position
Examples: Stacks, Queues
Value-oriented ADT:
Insert, delete, or ask questions about data items containing a specific value
Example?

4. Introduction & Terminology
Hierarchical (Tree) structure:
Parent-child relationship between elements of the structure
More?
Examples:
Organization chart
Library card-catalog
More???

5. Introduction & Terminology
Example: Library card catalog
Card Catalog
Subject Catalog
Author Catalog
Title Catalog
A - Abbex
Stafford , R - Stanford, R
Zon - Zz
Stafford, R. H.
Standish, T. A.
Stanford Research Institute

6. Introduction & Terminology
Table of contents of a book:

7. Trees and Tree Terminology
A tree is a data structure that consists of a set of nodes (vertex) and a set of edges (or branches) that connect nodes.

8. Introduction & Terminology
A is the root node.
B is the parent of D and E.
C is the sibling of B
D and E are the children of B
D, E, F, G, I are external nodes, or leaves
A, B, C, H are internal nodes
The depth (level) of E is 2
The height of the tree is (3)/4
The degree of node B is 2
Property?

9. Introduction & Terminology
Path from node n1 to nk:
A sequence of nodes n1, n2, … . nk such that there is an edge between each pair of nodes (n1, n2) (n2, n3), (nk-1, nk)
Height h:
The number of nodes on the longest path from the root to a leaf
Ancestor-descendent relationship
Ancestor of node n:
A node on the path from the root to n
Descendent of node n:
A node on the path from n to a leaf

10. Introduction & Terminology
Nodes:
Contains information of an element
Each node may contain one or more pointers to other nodes
A node can have more than one immediate successor (child) that lies directly below it
Each node has at most one predecessor (parent) that lies directly above it

11. Binary Trees
A binary tree is an ordered tree in which every node has at most two children.
A binary tree is:
either empty
or consists of a
root node
(internal node)
a left binary tree and
a right binary tree
Each node has at most two children
Left child
Right child

12. Binary Trees Special trees
Left (right) subtree of node n:
In a binary tree, the left (right) child (if any) of node n plus its descendants
Empty BT:
A binary tree with no nodes
Binary tree is position-oriented or value-oriented? But?

13. Number Of Nodes-height is h
Suppose the height is h
Minimum number of nodes in a binary tree:
At least one node at each of first h levels.
Maximum number of nodes in a binary tree:
All possible nodes at first h levels are present

14. Number Of Nodes & Height
Let n be the number of nodes in a binary tree whose height is h.
h <= n <= 2h – 1
If we know n, what will be the value for h??

15. Full Binary Trees In a full binary tree,
every leaf node has the same depth
every non-leaf node (internal node) has two children. A B D E H I J K C F G L A B C D E F G H I J K M N O L
Not a full binary tree.
A full binary tree.

16. Complete Binary Tree In a complete binary tree
every level except the deepest must contain as many nodes as possible ( that is, the tree that remains after the deepest level is removed must be full).
at the deepest level, all nodes are as far to the left as possible. A B D E H I J K C F G L A B D E H I J K C F G L
Not a Complete Binary Tree
A Complete Binary Tree

17. Proper Binary Tree In a proper binary tree,
each node has exactly zero or two children
each internal node has exactly two children. A B D E H I J K C F G L A B C D E F G H I J K
Not a proper binary tree
A proper binary tree

18. An Aside: A Representation for Complete Binary Trees
Since tree is complete, it maps nicely onto an array representation. A B D E H I J K C F G L
last T:
A B C D E F G H I J K L

19. Properties of the Array Representation
Data from the root node is always in T[0].
Suppose some node appears in T[i]
data for its parent is always at location T[(i-1)/2]
(using integer division)
data for it’s children nodes appears in locations
T[2*i+1] for the left child
T[2*i+2] for the right child
formulas provide an implicit representation of the edges
can use these formulas to implement efficient algorithms for traversing the tree and moving around in various ways.

20. What Tree? A B D E H I J K C F L

21. Balanced vs. Unbalanced Binary Trees
A binary tree in which the left and right subtrees of any node have heights that differ by at most 1
start A B C F G A B D E H I J K C F G L
root L D H I E
Mostly Unbalanced
Sort of Balanced J K

22. Binary search tree (BST)60 70 20 10 40 50 30
Binary search tree (BST)
A binary tree where
The value in any node n is greater than the value in every node in n’s left subtree
The value in any node n is less than the value of every node in n's right subtree
The subtrees are binary search trees too

23. Comparing Trees
These two trees are not the same tree! A B A B
Why?

24. Review Which of the following ADT is value-oriented? list sorted list
stack
queue
binary tree

25. Review The ______ is a position-oriented ADT that is not linear.
sorted list
queue
binary tree
list

26. Review
A node of a tree is called a(n) ______.
edge
root
branch
vertex

27. Review The lines between the nodes of a tree are called ______.
branches
edges
Arches
subtrees

28. Review
The node that is directly above node n in a tree is called the ______ of node n.
root
leaf
parent
child

29. Review In a tree, the children of the same parent are called ______.
leafs
siblings
roots
contemporaries

30. Review Each node in a tree has ______. exactly one parent
at most one parent
exactly two parents
at most two parents

31. Review
A descendant of node n is a node on a path from node n to ______.
the root
a leaf
a sibling of node n
a child of node n

32. Review A subtree of node n is a subtree rooted at ______. node n
the parent of node n
a child of node n
a sibling of node n

33. Review
The ______ of a tree is the number of nodes on the longest path from the root to a leaf.
height
length
width
age

34. Review
In a ______ of height h, all nodes that are at a level less than h have two children each.
general tree
binary tree
full binary tree
complete binary tree

35. Review
A ______ of height h is full down to level h – 1, with level h filled in from left to right.
full binary tree
complete binary tree
balanced binary tree
general tree

36. Review
In ______, the left and right subtrees of any node have heights that differ by at most 1.
all trees
all binary tress
n-ary trees
balanced binary trees

37. Review
Which of the following is NOT a property of a complete binary tree of height h?
all nodes at level h – 2 and above have two children each
when a node at level h – 1 has children, all nodes to its left at the same level have two children each
when a node at level h – 1 has one child, it is a left child
all leaves are at level h

38. Review
In an array based representation of a complete binary tree, which of the following represents the left child of node tree[i]?
tree[i+2]
tree[i–2]
tree[2*i+1]
tree[2*i+2]