 Consider a (2,4)tree that is
maximally populated with entries
(i.e., each internal node
has 3 keys (4 subtrees)
 The largest possible
number of entries stored in a
tree of height h is:
# nodes = 1 + 4 + 4^{2} + 4^{3} + ... + 4^{h1}
4^{h}  1
=  (Maple: sum( 4^k, k=0..(h1));)
3
Max. # keys = 3 * # nodes
= 4^{h}  1
Therefore:
n ≤ max. value for n (# keys)
≤ 4^{h}  1
<==> n+1 ≤ 4^{h}
<==> ^{2}log(n+1) ≤ h*log(4)
<==> ^{2}log(n+1) ≤ 2*h
<==> ½ ^{2}log(n+1) ≤ h ........ (1)

 Now consider a (2,4)tree that is
minimally populated with entries
(i.e., each internal node
has 1 keys (2 subtrees)
 The smallest possible
number of entries stored in a
tree of height h is:
# nodes = 1 + 2 + 2^{2} + 2^{3} + ... + 2^{h1}
= 2^{h}  1 (Maple: sum( 2^k, k=0..(h1));)
Min. # keys = 1 * # nodes
= 2^{h}  1
Therefore:
n ≥ min. value for n (# keys)
≥ 2^{h}  1
<==> n+1 ≥ 2^{h}
<==> ^{2}log(n+1) ≥ h*log(2)
<==> ^{2}log(n+1) ≥ h ........ (2)

 From Equations (1) and (2),
we conclude that:
^{2}log(n+1)
 ≤ h ≤ ^{2}log(n+1)
2

Therefore:
