# Master Theorem but with 2 different T's from two different recurrent trees

I am studying the Master Theorem, but there is a case in which I have no knowledge on how one should proceed. If it's given:

$$T_1(n)= 1$$ if n=1

$$T_1(n)= 4T_1(\frac n2) + \sqrt{3n}+1$$ otherwise.

For this case $$T_1(n) \in \Theta(n^2)$$

We then have:

$$T_2(n)= 1$$ if n=1

$$T_2(n)= 14T_2(\frac n4)+ T_1(n)=14T_2(\frac n4)+4T_1(\frac n2) + \sqrt{3n}+1$$ otherwise.

If I'd like to define a,b,c how do I proceed?

Is the number of nodes in a layer for the recurrence tree 18 ?

I am unfamiliar with how one proceeds in this case.

• From the first part you already know $T_1(n) = \theta(n^2)$ thus: $T_2(n) = 14T_2(n/4) + \theta(n^2)$. Now you can apply the Master Theorem. Commented May 2, 2022 at 0:22
• @plshelp You should post that as an answer. Commented May 2, 2022 at 0:43
• @plshelp That's what I did but there is a contradiction when I try to find the asymptotic behavior for this Commented May 2, 2022 at 8:06
• When solving the second equation, $T_1$ is already known (at least asymptotically).
– user16034
Commented May 2, 2022 at 20:09