7/29/2019 L-10 Trees.ppt

http://slidepdf.com/reader/full/l-10-treesppt 1/24

Data Structures (FALL-2011)

Trees

Thursday, September 09, 2004

7/29/2019 L-10 Trees.ppt

http://slidepdf.com/reader/full/l-10-treesppt 2/24

Data Structures (FALL-2011)

Trees: Basic terminology[1]

•  Hierarchical data structure

• Each position in the tree is called a node 

• The “top” of the tree is called the root 

• The nodes immediately below a node are called itschildren; nodes with no children are called leaves (orterminal nodes), or terminal nodes, and the nodeabove a given node is its parent (or father)

• A node x is ancestor of node y if x is father of y or 

father of some ancestor of y. y is called descendent of x. Ancestor of a node is its parent, grand parent or grand-grand parent or so on…. 

7/29/2019 L-10 Trees.ppt

http://slidepdf.com/reader/full/l-10-treesppt 3/24

Data Structures (FALL-2011)

Trees: Basic terminology[2]

•  Nodes with the same parent are siblings 

• A node plus the collection of nodes beneath it

is called a subtree 

• The number of nodes in the longest path from

the root to a leaf is the depth (or height) of the

tree

 – is the depth 2 or 3?

 – depends on the author 

7/29/2019 L-10 Trees.ppt

http://slidepdf.com/reader/full/l-10-treesppt 4/24

Data Structures (FALL-2011)

Root

 Node A

 Node H

 Node B

 Node G Node F Node E Node D Node C

 Node K  Node J

 Node L

 Node I

 Nodes C, H, and I form a subtree Nodes H and I are siblings

C is H’s parent, H and I are children of C 

What is the depth of this tree?

7/29/2019 L-10 Trees.ppt

http://slidepdf.com/reader/full/l-10-treesppt 5/24

Data Structures (FALL-2011)

Binary Trees

• A commonly used type of tree is a binary tree

• Each node has at most two children

• The “tree” is a conceptual structure 

• The data can be stored either in a dynamiclinked tree structure, or in contiguous

memory cells (array) according to a set  pattern; in other words, implementation can be pointer-based or array-based

7/29/2019 L-10 Trees.ppt

http://slidepdf.com/reader/full/l-10-treesppt 6/24

Data Structures (FALL-2011)

Binary trees

• A tree is called strictly binary tree if every non leaf 

node has exactly two children.

• A strictly binary tree having n leaves always contain

2n-1 nodes• Text book definition of depth of a tree

 –  Depth of binary tree is maximum level of any leaf in the

tree.

 –  Root of tree is at level 0, and level of any other node in the

tree is one more than the level of its father 

7/29/2019 L-10 Trees.ppt

http://slidepdf.com/reader/full/l-10-treesppt 7/24

Data Structures (FALL-2011)

Binary Trees

• A complete binary tree of level d is the strictly binary

tree all of whose leaves are at level d 

• A complete binary tree of level d has 2l  nodes at each

level l where 0<=l <=d• Total number of nodes (tn) in a complete binary tree of 

depth d will be:

1222222 1

0

210

d 

d 

  j

  jd tn

For a binary tree of height h there are O(2h) nodes

7/29/2019 L-10 Trees.ppt

http://slidepdf.com/reader/full/l-10-treesppt 8/24

Data Structures (FALL-2011)

Binary Trees

• For a complete binary tree if number of nodes (tn) areknown, then we may find depth of the binary tree

1-1)(log

1)(log1

12

12

2

2

1

1

tnd 

tnd 

tn

tn

d 

d 

A binary tree with nodes n has height O(log2n)

7/29/2019 L-10 Trees.ppt

http://slidepdf.com/reader/full/l-10-treesppt 9/24

Data Structures (FALL-2011)

Binary Trees Storage

• Linked List based implementation

• Array based implementation

7/29/2019 L-10 Trees.ppt

http://slidepdf.com/reader/full/l-10-treesppt 10/24

Data Structures (FALL-2011)

Binary Tree: as a linked structure

• Each node in the tree consists of:

 – The data, or value contained in the element

 – A left child pointer (pointer to first child)

 – A right child pointer (pointer to second child)

7/29/2019 L-10 Trees.ppt

http://slidepdf.com/reader/full/l-10-treesppt 11/24

Data Structures (FALL-2011)

Binary Tree: as a linked structure

• A root pointer points to the root node

 – Follow pointers to find every other element in the

tree

• Add and remove nodes by manipulating

 pointers

• Leaf nodes have child pointers set to null

7/29/2019 L-10 Trees.ppt

http://slidepdf.com/reader/full/l-10-treesppt 12/24

Data Structures (FALL-2011)

class CBinTree

{

struct Node

{

int value;Node *LeftChild,*RightChild;

}*Root;

/***Operations*********//************************/

};

7/29/2019 L-10 Trees.ppt

http://slidepdf.com/reader/full/l-10-treesppt 13/24

Data Structures (FALL-2011)

ROOT

7/29/2019 L-10 Trees.ppt

http://slidepdf.com/reader/full/l-10-treesppt 14/24

Data Structures (FALL-2011)

Binary Tree: contiguous storage

• Value in root node stored first, followed by left child,then right child

• Each successive level in the tree stored left to right;unused nodes in tree represented by a bit pattern to

indicate nothing stored there• Children of any given node n is stored in cells 2n and

2n + 1 (If array index starts at 1) 

• Can calculate position for any given node

• Storage allocated as for full tree, even if many nodesempty

• For a tree of depth h we need array of 2h+1-1 cells 

D S (FA 2011)

7/29/2019 L-10 Trees.ppt

http://slidepdf.com/reader/full/l-10-treesppt 15/24

Data Structures (FALL-2011)

Binary Trees

• Full binary tree of height (depth) h: all nodes

at a height less than h have exactly two

children• Balanced binary tree: for each node, the

difference in depth of the right and left

subtrees is no more than one• Completely balanced tree: left and right

subtrees of every node have the same height

D t St t (FALL 2011)

7/29/2019 L-10 Trees.ppt

http://slidepdf.com/reader/full/l-10-treesppt 16/24

Data Structures (FALL-2011)

Full Binary Tree

A very sparse,unbalanced tree

D t St t (FALL 2011)

7/29/2019 L-10 Trees.ppt

http://slidepdf.com/reader/full/l-10-treesppt 17/24

Data Structures (FALL-2011)

Primitive Operations

• Left(node): Gives index/pointer of left child

• Right(node): Gives index/pointer of right child

• Parent(node): Returns index/pointer of parent

• Brother(node): Returns index/pointer of brother 

• Root: Gives index/pointer of root node

• Info(Node): Data/Info stored at node

• IsLeft(node): Is node left child? Yes/No

• IsRight(node): Is node right child? Yes/No

D t St t (FALL 2011)

7/29/2019 L-10 Trees.ppt

http://slidepdf.com/reader/full/l-10-treesppt 18/24

Data Structures (FALL-2011)

Common Operations

• Tree traversal

•  Node addition

•  Node deletion• Destroy

D t St t (FALL 2011)

7/29/2019 L-10 Trees.ppt

http://slidepdf.com/reader/full/l-10-treesppt 19/24

Data Structures (FALL-2011)

Traversal of Binary Trees

• Pass through all nodes of tree

• Inorder (symmetric traversal)

• Preorder (depth first traversal)• Postorder 

Data Str ct res (FALL 2011)

7/29/2019 L-10 Trees.ppt

http://slidepdf.com/reader/full/l-10-treesppt 20/24

Data Structures (FALL-2011)

Trees Traversal

• Inorder 

 – (Left) Root (Right)

• Preorder  – Root (Left) (Right)

• Postorder 

 – (Left) (Right) Root

Root

Left Right

Data Structures (FALL 2011)

7/29/2019 L-10 Trees.ppt

http://slidepdf.com/reader/full/l-10-treesppt 21/24

Data Structures (FALL-2011)

Inorder Traversal

Left Root Right manner 

• Left + Right• [Left*Right]+[Left+Right]

• (A*B)+[(Left*Right)+E)

• (A*B)+[(C*D)+E]

+

* +

A B * E

C D

(A*B)+(C*D+E)

Data Structures (FALL 2011)

7/29/2019 L-10 Trees.ppt

http://slidepdf.com/reader/full/l-10-treesppt 22/24

Data Structures (FALL-2011)

Preorder Traversal

Root Left Right manner 

• + Left Right

• + [*Left Right] [+Left Right]• +(*AB) [+ *Left Right E]

• +*AB + *C D E

+

* +

A B * E

C D

Data Structures (FALL 2011)

7/29/2019 L-10 Trees.ppt

http://slidepdf.com/reader/full/l-10-treesppt 23/24

Data Structures (FALL-2011)

Postorder Traversal

Left Right Root manner 

• Left Right +• [Left Right *] [Left Right+] +

• (AB*) [Left Right * E + ]+

• (AB*) [C D * E + ]+• AB* C D * E + +

+

* +

A B * E

C D

Data Structures (FALL 2011)

7/29/2019 L-10 Trees.ppt

http://slidepdf.com/reader/full/l-10-treesppt 24/24

Data Structures (FALL-2011)

Binary Search Tree

• Elements are sorted:

• For any node n in a binary search tree:

 –  The value at n is greater than the values of any of the nodes

in its left subtree –  The value at n is less than the values of any of the nodes in

its right subtree

 –  Its left and right subtrees are binary search trees

• Need to define “greater than” and “less than” for thespecific data