CSE 3358 NOTE SET 5

Data Structures and Algorithms

Trees

Overcome the linear nature of a list. Allows for storage of hierarchical data

Recursive Definition of Tree:

• An empty structure is empty tree• If t1, …, tk are disjointed trees, then the structures

whose root has as its children the roots of t1, …, tk is also a tree

• Only structures generated by these two rules are trees

Tree Terminology

• Root

• Children

• Parent

• Ancestors

• Descendants

Tree Terminology

A

B C

K L

G H

JI

FED

• Height of C

• Path from A to J

• Length of Path from A to J

• Depth of F

Tree Terminology

20

10 15

18 19

16 17

98

765

• Children of 10

• Ancestors of 18

• Descendant 15

• Leaves of the tree

Generic Tree Traversals

Pre-order Work is done on children before parent is processed

Post-order Work is done on parent after children are processed

Binary Tree

Each node has 2 children at max. Left and Right Benefit w/ respect to efficiency of

algorithms Complete Binary Tree

All non-leaf nodes have both children

All leaves are at same level

Worst Case Binary Tree

Binary Search Tree

Binary Search Property:

For each node n of the tree, all values stores in its left subtree are less than the value v stored in n and all values stored in the right subtree are greater than v.

v

• No =. Only < and >. • Storing multiple copies of the same value is avoided.• Meaning of < and > depend on what the data payload of the

node is• Average Depth is O(lg N)

Implementation

Can be implemented as: Array

struct node { T* payload; int leftIdx; int rightIdx;};

If element deleted, update of entire tree may be necessary. Linked Dynamic structure

struct node { T* payload; node *left, *right};

Search Intuition

Complexity of Search

Based on the number of comparisons performed during the searching process depends on the number of nodes on the unique path

from root to the node being searched for. Complexity: The length of the path leading to this

node plus 1. depends on the overall shape of the tree

Think about a tree that is essentially a List vs a balanced tree

Internal Path Length:

The sum of all path lengths of all nodes.

IPL

14

12 16

IPL

14

7 27

3 9 32

IPL

14

7 27

3 9 32

Formula for IPL:

Formula for Average IPL:

IPL for Binary Tree

Worst Case: tree that looks like a Linked List

Best Case: all leaves in tree of height h are in at most two levels only nodes in the next to last level can have children

2

3

4

5

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