What is h in terms of n and t?

Theorem 19.1

Given n $\geq$ 1, t $\geq$ 2, B-Tree of height h and minimum degree t, and number of keys n,

Proof:

n $\geq$ minimum #nodes in tree of height h and minimum degree t

The minimum #nodes means root has one key (two children) and other nodes have t-1 (minimum) keys.

= 1 key at root +

2(t-1) at depth 1 +

2t(t-1) at depth 2 +

2t²(t-1) at depth 3 + ...

= 1 + = 1 + 2(t-1)

= 1 +

= 1 + 2(t^h - 1)

= 2 t^h - 1

n $\geq$ 2t^h - 1

2t^h$\leq$ n+1

h $\leq$