1. Trees

2. Trees Set of elements or nodes one node as root
every other node has a single predecessor (parent) and zero or more successors (children)
nodes without successors are called leaves
Nodes are connected by edges
The depth of a node is the number of edges on the path to the root
The height of the tree is the number of edges on the longest path from root to any leaf
A complete tree has all leaves at same depth, all other nodes have the same number of children

3. Example leaf leaf leaf leaf leaf leaf leaf leaf root
height of tree = 3
 edge
leaf
leaf
leaf
leaf
leaf
leaf
leaf
leaf

4. Questions What is the height of the tree? What is the depth of node n?
What is the depth of node p?
Is the tree complete? n p

5. Questions What is the height of the tree? What is the depth of node n?
What is the depth of node p?
What is the depth of node q? p q n

6. Subtrees
The subtree rooted at x is the tree of all descendants of x that has node x as root q p
subtree rooted at q
subtree rooted at p

7. Binary trees
A binary tree is a structure defined on a finite set of nodes that either
contains no nodes
is composed of three disjoint sets of nodes
a root node
a binary tree called its left subtree
a binary tree called its right subtree

8. Left and right subtrees
In a binary tree
the left subtree of x is the tree rooted at the left child of x
the right subtree of x is the tree rooted at the right child of x x
left subtree
right subtree

9. Binary trees p q w x y z
left subtree of root
right subtree of root

10. Binary trees p q w x y z subtree rooted at p subtree rooted at q
left subtree of p
right subtree of p
left subtree of q
right subtree of q
subtree rooted at p
subtree rooted at q

11. Binary trees p q w x y z

12. Questions In a complete binary tree of height 3,
how many nodes are in the tree?
how many leaves are in the tree?
In a complete binary tree of height 4,
How many nodes in a complete binary tree of height h?
How many leaves in a complete binary tree of height h?
How many nodes in a binary tree of height h?
How many leaves in a binary tree of height h?

13. Complete binary tree of height 3
15 nodes
8 leaves

14. Complete binary tree of height 4
31 nodes
16 leaves

15. Binary trees In a binary tree of height h
the total number of nodes is at most
…+2h = 2h+1 –1
the number of leaf nodes is at most 2h
If the tree is complete
there are exactly 2h+1 – 1 nodes
there are exactly 2h leaves
binary tree
complete binary tree

16. Decision trees A decision tree begins with a decision you need to make
start with an initial decision node
ask a question
Structure for investigating options and the possible outcomes of choosing those options
result of a decision can be another decision
outcome is a terminal node
Tree should have unique paths from the decision node to each of the terminal nodes.
help you to choose between several courses of action

17. A decision tree yes yes no yes no no He received the
Physics Prize in 1921.
yes
Would you like to
read about Einstein?
yes
Try the Medicine
Prize in 1962. no
Would you like to
read about a scientist?
Look up the Peace
Prize in 1991.
yes no
Would you prefer
a humanitarian?
Try the Prize in
Literature, 1970. no

18. A decision tree for insertion sort
a  b?
sort three elements
a,b,c
yes no
b  c?
a  c?
yes no
yes no
a  c?
b  c?
a  b  c
b  a  c
yes no
yes no
a  c  b
c  a  b
b  c  a
c  b  a

19. Insertion sort of 4 elements
a  b?
sort four elements
a,b,c,d
best case path
yes no
b  c?
a  c?
yes no
c  d? …
b  c?
worst case path no
a  d?
a  b  c  d … … no
b  d? no …
c  d?
d  c  b  a

20. Decision-tree model Decision trees are used to model comparison sorts
insertion sort, mergesort, quicksort
Each possible permutation of n elements must appear as a leaf in the tree
at least n! leaves
Worst-case number of comparisons = length of longest path from root to leaf
Insertion sort worst case:
number of comparisons: 0+1+ … + n-1 = ½ (n2-n)
Length of longest path in decision tree for 4 elements = ½ (42-4) = 6