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B
K
Prove properties by induction on k
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 20.1).
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.
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