Data Structures and Algorithms in Java[1].pdf - Fulvio Frisone

(10.3)
> 2 i · n(h − 2i).
That is, n(h) > 2 i · n(h − 2i), for any integer i, such that h − 2i ≥ 1. Since we already
know the values of n(1) and n(2), we pick i so that h − 2i is equal to either 1 or 2.
That is, we pick
.
By substituting the above value of i in formula 10.3, we obtain, for h ≥ 3,
.
(10.4)
By taking logarithms of both sides of formula 10.4, we obtain
log n(h) > h/2 − 1,
from which we get
h < 2logn(h) + 2,
(10.5)
which implies that an AVL tree storing n entries has height at most 2logn + 2.
By Proposition 10.2 and the analysis of binary search trees given in Section 10.1,
the operations find and findAll, in a dictionary implemented with an AVL tree,
run in time O(logn) and O(logn + s), respectively, where n is the number of entries
in the dictionary and s is the size of the collection returned. Of course, we still have
to show how to maintain the height-balance property after an insertion or removal.
10.2.1 Update Operations
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