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

Code Fragment 7.3:
Method depth written in Java.
The running time of algorithm depth(T, v) is O(d v ), where d v denotes the depth of
the node v in the tree T, because the algorithm performs a constant-time recursive
step for each ancestor of v. Thus, algorithm depth (T, v) runs in O(n) worst-case
time, where n is the total number of nodes of T, since a node of T may have depth n
− 1 in the worst case. Although such a running time is a function of the input size, it
is more accurate to characterize the running time in terms of the parameter d v , since
this parameter can be much smaller than n.
Height
The height of a node v in a tree T is also defined recursively:
• If v is an external node, then the height of v is 0
• Otherwise, the height of v is one plus the maximum height of a child of v.
The height of a nonempty tree T is the height of the root of T. For example, the
tree of Figure 7.2 has height 4. In addition, height can also be viewed as follows.
Proposition 7.4: The height of a nonempty tree T is equal to the maximum
depth of an external node of T.
We leave the justification of this fact to an exercise (R-7.6). We present here an
algorithm, height1, shown in Code Fragment 7.4 and implemented in Java in
Code Fragment 7.5, for computing the height of a nonempty tree T based on the
proposition above and the algorithm depth from Code Fragment 7.2.
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