Binary tree

The depth of a root is

0

ne<=

2^h which equivalent to h>=log2(ne)

Child

A node directly connected to another node when moving away from the root.

Parent

A node directly connected to another node when moving towards the root

Ancestor

A node reachable by repeated proceeding from child to parent

Descendant

A node reachable by repeated proceeding from parent to child

Internal node

A node with at least one child

Leaf

A node with no children

Binary tree

A tree in which each node can have at most 2 children

Use a dynamic link or pointer to represent

Binary tree

Create a node using a class

Class BinaryNode{ private: int element; BinaryNode *left, *right; Public: BinaryNode( int el=0, BinaryNode *lt=Null, Binarynode*rt=Null): element(el), left(lt), right(rt){} }

h

Height of a tree

To find the height of a tree

Ht=1+max(HL,Hr) where Hl is the height of the left tree and the HR is the height of the right tree

Depth of x =

Length of path from root to z

A binary try can give order in searching a element

Log n

ne

Number of external nodes (leaves)

Tree properties 1

Recursive data structure

Application of tree it can

Storing naturally hierarchical data

Tree property

The number of edges is equal to the number of nodes minus one edge= node -1

Root

The top node in a tree

To build a binary tree the first thing is to build

a node that is done in binary node class

node that have no children

are call leaves or external nodes. All other nodes are called internal nodes

The number of nodes at level k is

at most 2^k

every node (except the root) is

connected by an edge from exactly one other node

Edge or link

connection between one node to another

Root is the

distinguished node it is the ancestor of all nodes in tree

Tree can organize data

for quick search, insertion, deletion

There is a unique path

from the root to each node

Height of tree

height of root node

Tree are use to represent a

hierarchical relation between data

A binary tree is a tree

in which each node has exactly two subtrees: The left subtree and right subtree, either or both of which may be empty

A general rooted tree

is a set of nodes that has designated node called the root, from which zero or more subtrees descend

The height of a tree

is equal to the height of the root

Depth of a node in a tree

is the length of the path from the root to the node

Size of a node or degree of a node

is the number of descendants

Size of a tree

is the total number of nodes of this tree

the recursive definition of the binary tree

it is either empty or consists of a root, a left tree, and right tree.

each subtree

itself satisfies the definition of a tree

Height of a node is the

length of the longest path from a given node to the deepest left.

Each arrow is

link

h >=

log2(n+1)-1 since ne=(n+1)/2

A tree with n nodes must have

n-1 edges

N node =

n-1 edges

ne=

ni+1

h <=

ni<=(n-1)/2 since ni=(n-1)/2

an empty tree has

no nodes

Siblings

node with the same parent

Height of x

number of edges in longest path from x to a leaf

Depth

number of edges in path from root to x

ni

number of internal nodes

n

number of nodes

A tree is

set of node and directed edges(or Branches) connecting pairs of node

To find the size of a tree

st=1+Sl+Sr where sl is the sixe of the left tree and sr is the size of the right tree

When a node has no children

the corresponding member are null

sibling are node with

the same parent

each node has

three data

the middle of the node is

to hold data

the left of the node is

to store the address of the left child

the right of the cell is

to store the address of the right child

A node representation has

two Binarynode member, one for the left child and one for the right child and data object type

If there is a path from the node u to node v then

u is ancestor of v and v is a descendant of u