## What is the minimal degree $d$ required for a B tree with $44*10^6$ keys so that it’s height is less than or equal to $5$

What is the minimal degree $$d$$ required so a B – tree with $$44*10^6$$ keys will have a height $$h$$, such that $$h\leq 5$$

My attempt was to build the tallest tree possible with minimum degree $$d$$ and $$n = 44,000,000$$ keys and then solve for $$d$$. That would mean any other tree with a minimal degree $$d’$$ such that $$d’\geq d$$ and $$n$$ keys will be shorter than the one I built:

at depth 0 , we have the root and that’s $$1$$ node

at depth 1, we got exactly $$2$$ nodes

at depth 2, since we’re going for the tallest tree each node will have a minimal number of keys so $$d-1$$ keys each, that means $$d$$ children each so a total of $$2d$$ nodes.

at depth 3, following the same reasoning , $$2d^2$$ nodes.

at depth $$h$$, there are $$2d^{h-1}$$ nodes

total number of keys is :

$$n = 1+ (d-1)\sum_{k=0}^{h-1} {2d^k} = 1 + (d-1) \frac{2(d^h-1)}{d-1} = 2d^h-1 = 44*10^6$$

so:

$$2d^5-1=44,000,000$$

$$d= 29.4$$

$$d\geq 30$$

is that even correct ?