**Theorem 19.1**

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

**Proof:**

n 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 +

2*t*^{2}(t-1) at depth 3 + ...

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

= 1 +

= 1 +
2(*t*^{h} - 1)

= 2 *t*^{h} - 1

n 2*t*^{h} - 1

2*t*^{h} n+1

h