**Base Case:** Holds for *B*_{0}.

**Assume:** Holds for
.

- 1.
*B*_{k}is 2 copies of*B*_{k-1}, so nodes.- 2.
- Depth of
*B*_{k}is one greater than maximum depth of*B*_{k-1}.

Add one more level: height = (k-1) + 1 = k. - 3.
- See book (Lemma 19.1).
- 4.
- True for children
*B*_{k-1},*B*_{k-2}, ..,*B*_{0}from left to right.

*B*_{k-1}is left child of*B*_{k}, root is also root of*B*_{k-1}(minus left child), so degrees are*B*_{k-1},*B*_{k-2}, ..,*B*_{0}.

The root of*B*_{k}is a*B*_{k-1}with one more child (the left child), so root of*B*_{k}has degree (k-1) + 1 = k.