a$)$ Consider the following function:

Tree$-$Search $(x,k)$

Write the recurrence relation for TREE$-$SEARCH.

b$)$ Give an asymptotic solution for the following recurrence:

$T(n)=4T(\frac{n}{3})+n^2,$ with $T(1)=1$ using the tree method.

i$.$ Draw the first $3$ levels of the tree.

ii$.$ What is the input size at level $6?$

iii. What is the complexity at level $5?$

$[2$ marks$]$

iv$.$ Find the Total complexity of the tree.

c$)$ Consider the recurrence $T(n)=3T(\frac{n}{3})+n,$ with $T(1)=1$

Use the substitution method to solve this recurrence and find the asymptotic upper

bound. Do not forget to write down precisely what you are trying to prove and to

include the base case, inductive hypothesis, and inductive step.