# Max no. of keys of B-Tree of height h

I am reading about B-Tree through CLRS. One of the exercise problem is to express maximum number of keys in a BTree of height $$h$$ and minimum degree $$t$$.

Properties of BTree of $$t$$:

• $$t\ge 2$$
• Min. no of keys in a node is $$t-1$$ and max. number of keys in a node is $$2t-1$$
• Max number of children a node can have is $$2t$$

To solve it the formula I came up with is $$\sum_0^h (2t)^h(2t-1)$$

Does it have any convergence?

• convergence doesn't seem to be the precise term. Feb 2 '20 at 7:52
• @greybeard that was the closes term that I could think of. Maybe reduction??
– Sab
Feb 2 '20 at 7:59

$$\sum_{i=0}^h (2t)^i(2t-1) = (2t-1) \sum_{i=0}^h (2t)^i = (2t-1) \frac{(2t^{h+1}-1)}{(2t-1)} = (2t)^{h+1}-1$$