The given recurrence relation is:

$T(n) = T\left(\frac{n}{10}\right) + T\left(\frac{9n}{10}\right) + n$

Step 1: Analyze the Recurrence Relation

This is a non-homogeneous recurrence relation that combines two subproblems of sizes $\frac{n}{10}$ and $\frac{9n}{10}$, plus a linear cost $n$.

Step 2: Apply the Master Theorem (if applicable)

The recurrence has the form:

$T(n) = a_1 T\left(\frac{n}{b_1}\right) + a_2 T\left(\frac{n}{b_2}\right) + f(n)$

For our case:

• $a_1 = 1$, $b_1 = 10$
• $a_2 = 1$, $b_2 = \frac{10}{9}$
• $f(n) = n$

This doesn't directly match the standard form for the Master Theorem, so we'll use a different approach.

Step 3: Use the Recursion Tree Method

To get a better understanding, let's construct a recursion tree for this recurrence relation.

• At the first level, the cost is $n$.
• At the second level, the cost is $\frac{n}{10}$ from the first subproblem and $\frac{9n}{10}$ from the second subproblem, giving a total cost of $n$.
• At the third level, the costs further split as $\frac{n}{100}, \frac{9n}{100}, \frac{81n}{100}, \ldots$, again summing to $n$.

The sum of costs at each level is $n$, and the recursion continues until the size of the subproblems becomes very small, i.e., when $n = 1$.

Step 4: Determine the Height of the Tree

The height of the tree is determined by the largest of the two subproblems. Since the larger subproblem is $\frac{9n}{10}$, the height of the tree is approximately:

$\text{Height} = \log_{\frac{10}{9}} n$

Step 5: Compute the Total Cost

The total cost of the recurrence is the sum of the costs at each level. Since the cost at each level is $n$, and there are $\log_{\frac{10}{9}} n$ levels, the total cost $T(n)$ is:

$T(n) = n \cdot \log_{\frac{10}{9}} n$

Using logarithm change of base formula $\log_{\frac{10}{9}} n = \frac{\log n}{\log \frac{10}{9}}$, the final solution for the recurrence relation is:

$T(n) = \Theta(n \log n)$

Final Answer:

The time complexity of the given recurrence relation is $\boxed{\Theta(n \log n)}$.

Would you like more details on any step or have any questions? Here are some related questions you may explore:

1. How does the recursion tree method work in solving recurrence relations?
2. What are the limitations of the Master Theorem?
3. Can you solve the same recurrence relation using substitution?
4. How would the complexity change if the linear term $n$ was different?
5. What are the implications of having different base cases in the recurrence?

Tip: Understanding the recursion tree method can be especially useful for solving complex recurrence relations that don’t fit the standard forms.