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

Code Fragment 7.6: Algorithm height2 for
computing the height of the subtree of tree T rooted
at a node v.
Code Fragment 7.7:
Java.
Method height2 written in
Algorithm height2 is more efficient than height1 (from Code Fragment 7.4).
The algorithm is recursive, and, if it is initially called on the root of T, it will
eventually be called on each node of T. Thus, we can determine the running time
of this method by summing, over all the nodes, the amount of time spent at each
node (on the nonrecursive part). Processing each node in children(v) takes O(c v )
time, where c v denotes the number of children of node v. Also, the while loop
has c v iterations and each iteration of the loop takes O(1) time plus the time for
the recursive call on a child of v. Thus, algorithm height2 spends O(1 + c v )
time at each node v, and its running time is O(Σ v (1 + c v )). In order to complete the
analysis, we make use of the following property.
Proposition 7.5: Let T be a tree with n nodes, and let c v denote the
number of children of a node v of T. Then, summing over the vertices in T, Σ v c v =
n − 1.
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