After using the scheme for some time, Sally finds out that it has some problems and has asked for your help. To assist,

you will need to prepare answers to the following four questions:

$1.$ The main problem is that as Sally's circle of friends grows, it sometimes takes a long time to get a response to

SEARCH calls, which is using up valuable talk$-$time on Sally$$s mobile phone plan.

$$ Explain the circumstances that would result in unnecessarily long SEARCH calls and what can be done to

minimise the problem.

$$ If on average a SEARCH call takes one minute $$ deciding who to call, placing the call, and reporting its

outcome $$ what is the maximum time that a SEARCH might take if Sally added all $180$ of her friends to

her BST$?$

$$ If the problem could be fixed, how much time would a typical SEARCH take?