Do I need to purchase a seat for my child?

In this particular case, the answer is yes you will need an extra seat.

Generally the rules for most airlines are as follows - however we suggest double checking with your travel agent or airline at the time of booking to confirm the booking regulations.

International Flights:
Infants who do not occupy a separate seat are carried for 10% of the applicable adult fare.
Each infant under 2 years must be accompanied by a separate adult.
Children over 2 years must occupy a separate seat.
All children and infants must have their own tickets, whether or not they are occupying a separate seat.

Domestic Flights:
Children under 2 years of age who do not occupy a seat are carried free of charge.
Each infant must be accompanied by a separate adult.
Children over 2 must occupy a separate seat.
Children under 2 can travel on their parents tickets unless a separate seat has been booked and a fare paid.