1 trees a tree is a data structure used to represent different kinds of data and help solve a number...

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Trees

• A tree is a data structure used to represent different kinds of data and help solve a number of algorithmic problems

• Game trees (i.e., chess ), UNIX directory trees, sorting trees etc

• We will study extensively two useful kind of trees: Binary Search Trees and Heaps

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Trees: Definitions

• Trees have nodes. They also have edges that connect the nodes. Between two nodes there is always only one path.

Tree nodes

Tree edges

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Trees: More Definitions

• Trees are rooted. Once the root is defined (by the user) all nodes have a specific level.

• Trees have internal nodes and leaves. Every node (except

the root) has a parent and it also has zero or more children.

level 0

level 1

level 2

level 3

root

internal nodes

leaves

parentandchild

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Binary Trees

• A binary tree is one that each node has at most 2 children

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Binary Trees: Definitions

• Nodes in trees can contain keys (letters, numbers, etc)• Complete binary tree: the one where leaves are only in

last two bottom levels with bottom one placed as far left as possible

14

10

15128

16

7 9 11 13

18

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Binary Trees: Array Representation

• Complete Binary Trees can be represented in memory with the use of an array A so that all nodes can be accessed in O(1) time:

– Label nodes sequentially top-to-bottom and left-to-right

– Left child of A[i] is at position A[2i]– Right child of A[i] is at position A[2i + 1]– Parent of A[i] is at A[i/2]

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Binary Trees: Array Representation

14

10

15128

16

7 9 11 13

18

1

2 3

4 5 6 7

8 9 10 11

1 2 3 4 5 6 7 8 9 10 11

14 10 16 8 12 15 18 7 9 11 13Array A:

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Binary Trees: Pointer Representation

• To freely move up and down the tree we need pointers to parent (NIL for root) and pointers to children (NIL for leaves)

typedef tree_node {int key;struct tree_node *parent;struct tree_node *left, *right;

}

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Tree Traversal: InOrder

InOrder is easily described recursively:

•Visit left subtree (if there is one) In Order•Visit root•Visit right subtree (if there is one) In Order

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Another common traversal is PreOrder.It goes as deep as possible (visiting as it goes) then left to right.

More precisely (recursively):•Visit root•Visit left subtree in PreOrder•Visit right subtree in PreOrder

Tree Traversal: PreOrder

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PreOrder can also be implemented with a stack, without recursion:

Stack Spush root onto Srepeat until S is empty

v = pop Sif v is not NULL

visit vpush v’s right child onto Spush v’s left child onto S

Tree Traversal: Non-recursive

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PostOrder traversal also goes as deep as possible, but only visitsinternal nodes during backtracking.

More precisely (recursive):•Visit left subtree in PostOrder•Visit right subtree in PostOrder•Visit root

Tree Traversal: PostOrder

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Tree Traversal: LevelOrder

LevelOrder visits nodes level by level from root node:

• Can be implemented easily with a queue (how??)• We will learn this is called Breadth First Search in the data structure called graphs later in 242

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Binary Search Trees

• A Binary Search Tree (BST) is a binary tree with the following properties:

– The key of a node is always greater than the

keys of the nodes in its left subtree

– The key of a node is always smaller than the

keys of the nodes in its right subtree

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Binary Search Trees: Examples

C

A D

14

10

15118 18

16

14

10

15118

16

root

root

root

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Binary Search Trees

• BST is a tree with the following BST property:

key.left < key.parent < key.right

NOTE! When nodes of a BST are traversed by Inorder traversal the keys appear in sorted order:

inorder(root) { inorder(root.left) print(root.key) inorder(root.right)}

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Binary Search Trees: Inorder

14

10

15118 18

16

print(8)print(10)print(11)print(14)print(15)print(16)print(18)

Inorder visits and prints:

Example:

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Searching for a key in a BST

Picture BST’s Parent and Left/Right subtrees as follows:

Problem: how do you search BST for node with key x ?

L

P

R

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search(root, x)

compare x to key of root:

- if x = key return - if x < key => search in L - if x > key => search in R

search L or R in the exact same way

Example:

14

10

15118

16

x=8 is ??? X=17 is ???

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searchBST(root, key) /* found init to false */

if root=nil return

if root.key=key then

found=true

return root

else if key < root.key then

searchBST(root.L, key)

else

searchBST(root.R, key)

How can you rewrite searchBST non-recursively?

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Inserting a new key in a BST

How to insert a new key?

The same procedure used for search also applies: Determine the location by searching. Search will fail. Insert new key where the search failed.

Example:

10

8

5

3 4

2

9

Insert 4?

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Building a BST

Build a BST from a sequence of nodes read one a time

Example: Inserting C A B L M (in this order!)

1) Insert C

C

2) Insert A

C

A

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Building a BST

3) Insert B C

A

B

4) Insert L

5) Insert M

C

A

B

L

MC

A

B

L

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Building a BST

Is there a unique BST for letters A B C L M ?

NO! Different input sequences result in different trees

C

A

B

L

M

A

B

C

L

M

Inserting: A B C L M Inserting: C A B L M

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Sorting with a BST

Given a BST can you output its keys in sorted order?

Visit keys with Inorder: - visit left

- print root - visit right

How can you find the minimum? How can you find the maximum?

C

A

B

L

M

Example:

A B C L M

Inorder visit prints:

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Deleting from a BST

To delete node with key x first you need to search for it. Once found, apply one of the following three cases

CASE A: x is a leaf

x

delete x BST property maintained

q

r

q

r

p p

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Deleting from a BST cont.

Case B: x is interior with only one subtree

L

x

q

r

delete x L

q

r

BST property maintained

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Case C: x is interior with two subtrees

W

x

q

r

delete x

Z

t s

r

W

q

r

delete x

s Z

r

BST property maintained

t

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Case C cont: … or you can also do it like this

W

q

r

Zs

r

t

q < x < r

Q is smaller than the smaller element in Z R is larger than the largest element in W

Can you see other possible ways ?

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Complexity of Searching with BST

• What is the complexity of searchBST ?

• It depends on:– the key x– The other data– The shape of the tree (full, arbitrary)

Complexity Analysis: We are interested in best case, worst case and average case

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level 0

level 1

level 2

level 3

height (or depth) of tree = 1 + maximum_level

(1+3=4)

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If all nodes in the tree existthen it is called a full BST

If all levels are full except for the last level then it iscalled minimum-level BST

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Theorem: A full BST of height h has 2 - 1 nodes

Proof: Use induction

Inductive Basis: A tree of height 1 has one node (root)

Inductive Hypothesis: Assume that a tree of height h has

2 - 1 nodes

h

h

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Inductive step: Connect two trees of height h to make one of height h+1. You need to add one more node for the new root

root

L Rh

h+1

By inductive hypothesis the new number of nodes is

(2 - 1) + (2 -1) + 1 = 2 - 1 ……proved!h h h+1

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Theorem: For a minimum level tree of height h:

2 - 1 < # of nodes < 2 - 1hh - 1

Corollary: The tree with the smallest height with n # of nodes it has a height of h = 1 + log n

n 2 WHY?

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Therefore, for a full BST with N nodes the following holds for searchBST:

best time analysis ………… O(1)

worst time analysis ………… O(log N)

average case analysis ………… ???

We need to find what the average is!!


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• To define what average means you need to:– Find out all possibilities and …– Determine the probability of each possibility …– that is, you need to find the expected value:


n

j

h

j

xxE1

jxx Pr

Possible values for the number of steps j are 1, 2, …, h as we assume that key search value is equally to occur

1 trees a tree is a data structure used to represent different kinds of data and help solve a number...

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