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

h≤log(n+1) (10.10)
are true.
To justify these claims note first that, by the size property, we can have at most 4
nodes at depth 1, at most 4 2 nodes at depth 2, and so on. Thus, the number of
external nodes in T is at most 4 h . Likewise, by the depth property and the
definition of a (2,4) tree, we must have at least 2 nodes at depth 1, at least 2 2
nodes at depth 2, and so on. Thus, the number of external nodes in T is at least 2 h
In addition, by Proposition 10.7, the number of external nodes in T is n+ 1.
Therefore, we obtain
2 h ≤n+1
and
n+1≤4 h .
Taking the logarithm in base 2 of each of the above terms, we get that
h≤log(n+1)
and
log(n+1) ≤2h,
which justifies our claims (10.9 and 10.10).
Proposition 10.8 states that the size and depth properties are sufficient for keeping
a multi-way tree balanced (Section 10.4.1). Moreover, this proposition implies
that performing a search in a (2,4) tree takes O(logn) time and that the specific
realization of the secondary structures at the nodes is not a crucial design choice,
since the maximum number of children d max is a constant (4). We can, for
example, use a simple ordered dictionary implementation, such as an array-list
search table, for each secondary structure.
10.4.2 Update Operations for (2,4) Trees
Maintaining the size and depth properties requires some effort after performing
insertions and removals in a (2,4) tree, however. We discuss these operations next.
Insertion
To insert a new entry (k,x), with key k, into a (2,4) tree T, we first perform a
search for k. Assuming that T has no entry with key k, this search terminates
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