Binary Search Trees - Data Structures - Lecture Slides, Slides of Data Structures and Algorithms

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Binary Search Trees

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Tree Data Structure

•^

A non-linear data structure that follows theshape of a tree (i.e. root, children, branches,leaves etc)

-^

Most applications require dealing withhierarchical data (eg: organizational structure)

-^

Trees allows us to find things efficiently^ – Navigation is O(log n) for a “balanced” tree with n

nodes

•^

A Binary Search Tree (BST) is a data structurethat can be traversed / searched according to anorder

-^

A binary tree is a tree such that each node canhave at most 2 children.

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Definition of Flat T

-^

Given a Binary Tree T, Flat(T) is asequence obtained by traversing the treeusing

inorder

traversal

• That is for each node, recursively visit -^

Left Tree, Then Root, then Right Tree

root

L

R

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Flattening a BT

a

b

d

e

f^

g

T

flat(T) = e, b,f,a,d,g

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BT Definitions

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Definitions

• A

path

from node n

1

to n

k^

is defined as a

path n

,n 1

, …. n 2

k^

such that n

is the parenti

of n

i+

• Depth of a node

is the length of the path

from root to the node.

-^

Height of a node

is length of a path from

node to the deepest leaf.

• Height of the tree

is the height of the root

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Binary Tree Questions

•^

What is the

maximum height

of a binary tree

with n nodes? What is the

minimum height

•^

What is the minimum and maximum

number of

nodes

in a binary tree of height h?

•^

What is the

minimum number

of nodes in a full

tree of height h?

-^

Is a complete tree a full tree?

-^

Is perfect tree a full and complete tree?

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Binary Tree Properties

-^

Counting Nodes in a Binary Tree^ – The max number of nodes at level

i^

is 2

i^ (i=0,1,…,h)

• Therefore total nodes in all levels is–^

Find a relation between

n

and

h.

-^

A

complete tree

of height,

h

, has between 2

h^

and 2

h +

nodes.

-^

A

perfect

tree of height h has 2

h +

-1 nodes

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BST Operations

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Tree Operations

•^

Tree Traversals^ –

Inorder, PreOrder, PostOrder

Level Order

•^

Insert Node, Delete Node, Find Node

-^

Order Statistics for BST’s^ –

Find k

th^

largest element

num nodes between two values

•^

Other operations^ – Count nodes, height of a node, height of a tree,

balanced info

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Inorder Traversal

private void inorder(BinaryNode root){

if (root != null) {

inorder(root.left); process root; inorder(root.right); }

}

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Preorder Traversal

private void preorder(BinaryNode root){

if (root != null) {

process root; preorder(root.left);preorder(root.right); }

}

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Level order or Breadth-first traversal •Visit nodes by levels• Root is at level zero• At each level visit nodesfrom left to right• Called “

Breadth-First-

Traversal(BFS)”

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Level order or Breadth-first traversal

enqueue the rootwhile (the queue is not empty){

dequeue the front elementprint itenqueue its left child (if present)enqueue its right child (if present)

BFS Algorithm