Introduction to Trees

Linear Lists and Trees

• Linear lists (arrays, linked list)are useful for serially ordered data– (e1,e2,e3,…,en)

– Days of week– Months in a year– Students in a class

• Trees are useful for hierarchically ordered data– directory structure– Parenthetical Listing

– family trees:• all descendants of a particular person• all ancestors born after year 1800 of a particular person

Parenthetical Listing

A (B (C D) E F (G H I) )

Tree A tree is a collection of Nodes and a finite set of directed lines called branches, that connect the nodes. Each node can have 0 or more children A node can have at most one parent Tree defined recursively A tree consists of a distinguished node r, called the root, and zero or more

non-empty (sub) trees T1, T2, …, Tk each of whose roots are connected by a directed edge from r.

A tree is a collection of N nodes, one of which is the root and N-1 edges. The number of branches associated with a node is the degree of the

node.

Binary tree Tree with 0–2 children per node

Terminology Root no parent Leaf no child Interior non-leaf Height distance from root to leaf The root of each sub tree is said to be a child of r and r is

said to be the parent of each sub tree root. Siblings nodes with the same parent

• Depth (of node): the length of the unique path from the root to a node.

• Depth (of tree): The depth of a tree is equal to the depth of its deepest leaf.

• Height (of node): the length of the longest path from a node to a leaf.– All leaves have a height of 0– The height of the root is equal to the depth of the tree

• When the branch is directed toward the node, it is indegree branch.

• When the branch is directed away from the node, it is an outdegree branch.

• The sum of the indegree and outdegree branches is the degree of the node.

Recursive definition of a tree

• A tree is a set of nodes that either:- is empty or- has a designated node, called the root, from which hierarchically descend zero or more sub trees, which are also trees.

9

Binary trees

• A binary tree is a tree in which no node can have more than two subtrees; the maximum outdegree for a node is two.

• In other words, a node can have zero, one, or two subtrees.

• These subtrees are designated as the left subtree and the right subtree.

Picture of a binary treea

b c

d e

g h i

l

f

j k

Tree traversals

Preorder traversal• In preorder, the root is visited first• Here’s a preorder traversal to print out all

the elements in the binary tree:

A

B C

D E F G

A B D E C F G

Example1

Example2

Inorder traversal• In inorder, the root is visited in the middle• Here’s an inorder traversal to print out all

the elements in the binary tree:

Example1

Example1

Example2

Postorder traversal

• In Postorder, the root is visited last• Here’s a Postorder traversal to print out all

the elements in the binary tree:

Example1

Example1

Example2

Arithmetic Expression Tree• Binary tree for an arithmetic expression

– internal nodes: operators– leaves: operands

• Example: arithmetic expression tree for the expression ((2 (5 - 1)) + (3 2))

+

-2

5 1

3 2

Write down prefix, Infix & Postfix expression

Write down prefix, Infix & Postfix expression

Binary Search Trees

• Binary Search Tree Property: – the value stored at the root of a sub tree is greater than any value in its left sub tree

and less than any value in its right sub tree!

• Where is the smallest element?– leftmost element

• Where is the largest element?– rightmost element

• How to build & maintain binary trees?– Insertion– Deletion

• Maintain key property (invariant)– Smaller values in left sub tree– Larger values in right sub tree

Binary Search Trees• Key property

– Value at node• Smaller values in left subtree• Larger values in right subtree

– Example• X > Y• X < Z

Y

X

Z

Binary Search Trees

• Examples

Binary search trees

Not a binary search tree

5

10

30

2 25 45

5

10

45

2 25 30

5

10

30

2

25

45

Binary Search Tree – Insertion

• Algorithm1. Perform search for value X

2. Search will end at node Y (if X not in tree) 3. If X < Y, insert new leaf X as new left sub tree for Y 4. If X > Y, insert new leaf X as new right sub tree for Y

• Observations– O( log(n) ) operation for balanced tree

Example Insertion

• Insert ( 20 )

Binary Search Tree – Deletion

Algorithm

1. Perform search for value X

2. If X is a leaf, delete X

3. Else // must delete internal nodea) Replace with largest value Y on left sub tree OR smallest value Z on right sub treeb) Delete replacement value (Y or Z) from sub tree

Observation

- O( log(n) ) operation for balanced tree

Example Deletion (Leaf)

• Delete ( 25 )

5

10

30

2 25 45

10 < 25, right

30 > 25, left

25 = 25, delete

5

10

30

2 45

Example Deletion (Internal Node)

• Delete ( 10 )

5

10

30

2 25 45

5

5

30

2 25 45

2

5

30

2 25 45

Replacing 10 with largest

value in left sub tree

Replacing 5 with largest value in

left sub tree

Deleting leaf

Example Deletion (Internal Node)

• Delete ( 10 )

5

10

30

2 25 45

5

25

30

2 25 45

5

25

30

2 45

Replacing 10 with smallest value in right

sub tree

Deleting leaf Resulting tree

Iterative Search of Binary Tree

Node *Find( Node *n, int key) {

while (n != NULL) { if (n->data == key) // Found it

return n;

if (n->data > key) // In left subtree n = n->left;

else // In right subtree n = n->right;

} return null;}

Node * n = Find( root, 5);

Recursive Search of Binary TreeNode *Find( Node *n, int key) {

if (n == NULL) // Not foundreturn( n );

else if (n->data == key) // Found itreturn( n );

else if (n->data > key) // In left subtreereturn Find( n->left, key );

else // In right subtreereturn Find( n->right, key );

}

Node * n = Find( root, 5);

Example Binary Searches

• Find ( root, 2 )

5

10

30

2 25 45

5

10

30

2

25

45

10 > 2, left

5 > 2, left

2 = 2, found

5 > 2, left

2 = 2, found

root

Example Binary Searches

• Find (root, 25 )

5

10

30

2 25 45

5

10

30

2

25

45

10 < 25, right

30 > 25, left

25 = 25, found

5 < 25, right

45 > 25, left

30 > 25, left

10 < 25, right

25 = 25, found

Types of Binary Trees

• Degenerate – only one child• Balanced – “mostly” two children • Complete – always two children

Degenerate binary tree

Balanced binary tree

Complete binary tree

Introduction to Graph

• What is a graph?– A data structure that consists of a set of nodes (vertices)

and a set of edges that relate the nodes to each other

• The set of edges describes relationships among the vertices

• A graph G is defined as follows:G=(V,E)

V(G): a finite, nonempty set of verticesE(G): a set of edges (pairs of vertices)

Directed vs. undirected graphs

• When the edges in a graph have no direction, the graph is called undirected

Directed vs. undirected graphs (cont.)

• When the edges in a graph have a direction, the graph is called directed (or digraph)

Warning: if the graph is directed, the order of the vertices in each edge is important !!

Trees vs graphs

Trees are special cases of graphs!!

Graph terminology• Adjacent nodes: two nodes are adjacent if they are connected

by an edge– 5 is adjacent to 7– 7 is adjacent from 5

• Path: a sequence of vertices that connect two nodes in a graph

• Complete graph: a graph in which every vertex is directly connected to every other vertex

Graph terminology

• What is the number of edges in a complete directed graph with N vertices?

N * (N-1)

Graph terminology

• What is the number of edges in a complete undirected graph with N vertices?

N * (N-1) / 2

Graph terminology• Weighted graph: a graph in which each edge

carries a value

Graph implementation

• Array-based implementation– A 1D array is used to represent the vertices– A 2D array (adjacency matrix) is used to represent

the edges