Binary tree

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Binary Trees

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Tree Terminology

• A tree consists of a collection of elements or nodes, with each node linked to its successors

• The node at the top of a tree is called its root• The links from a node to its successors are

called branches• The successors of a node are called its children• The predecessor of a node is called its parent

Tree Terminology (continued)

• Each node in a tree has exactly one parent except for the root node, which has no parent

• Nodes that have the same parent are siblings• A node that has no children is called a leaf

node

Binary Trees

•A tree in which no node can have more than two children.

Example: Expression Trees

• Leaves are operands (constants or variables)• The other nodes (internal nodes) contain

operators• Will not be a binary tree if some operators are

not binary

Example: Expression Trees

Expression tree for ( a + b * c ) + ( ( d * e + f ) * g

Binary Tree Traversal

• Traversal is the process of visiting every node once

• 3 types of traversals*Inorder*Preorder*Postorder

Preorder traversal

Expression tree for ( a + b * c ) + (( d * e + f ) * g

• root, left, right• prefix expression

++a*bc*+*defg

Inorder traversal

Expression tree for ( a + b * c ) +( ( d * e + f ) * g

• left, root, right• infix expression

a+b*c+d*e+f*g

Postorder traversal

Expression tree for ( a + b * c ) + (( d * e + f ) * g

• left, right, root• postfix expression

abc*+de*f+g*+

InsertProceed down the tree as you would find a close matchIf X is found, do nothing (or update something)Otherwise, insert X at the last spot on the path traversed

DeleteConsider children of deleted nodeProperty of the search tree should be maintained.Three cases:(1) the node is a leaf

– Delete it immediately(2) the node has one child

– Adjust a pointer from the parent to bypass that node

Delete(3) the node has 2 children

– replace the key of that node with the minimum element at the right subtree

– delete the minimum element • Has either no child or only right child because if it has a left child, that

left child would be smaller and would have been chosen. So invoke case 1 or 2.

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