STACS 1991: STACS 91 pp 478-487

# Tight bounds on the path length of binary trees

• Alfredo De Santis
• Giuseppe Persiano
Algorithms II
Part of the Lecture Notes in Computer Science book series (LNCS, volume 480)

## Abstract

The external path length of a tree T is the sum of the lengths of the paths from the root to the external nodes. The maximal path length difference Δ is the difference of the lengths of the longest and shortest such path.

The external path length of binary trees with a given maximal path length difference Δ and given number of external nodes N has been studied by Klein and Wood. Namely, they have given upper bounds by using some results in [5] concerning properties of the ratio of the geometric and the harmonic means of integers (see [1]) and Lagrange multipliers (see [2]).

In this paper, we develop a new and very simple technique to obtain upper bounds. This allows us to present a simple derivation of their upper bound and successively improve their result. Namely, we derive a more precise upper bound that is also tight for every Δ and infinitely many N. We also manage to characterize for each N the tree with longest path length and Δ=2 and thus derive a matching upper bound for the case Δ=2; i.e. a bound that is achieved for all N. Finally, we initiate the study of lower bounds by presenting a matching lower bound for the case Δ=2.

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### References

1. [1]
R. Klein and D. Wood, “On the Path Length of Binary Trees”, Journal of the ACM, vol. 36, n. 2, April 1989, pp. 280–289.
2. [2]
R. Klein and D. Wood, “On the Path Length of Binary Trees”, Information Processing 89, Proceedings of the IFIP 11th World Computer Congress, San Francisco, USA, August 28–September 1, 1989.Google Scholar
3. [3]
D. E. Knuth, “The Art of Computer Programming”, vol. 3 “Sorting and Searching”, Addison-Wesley, reading, Mass., 1973.Google Scholar
4. [4]
J. Nievergelt and C. K. Wong, “Upper Bounds for the Total Path length of Binary Trees”, J. ACM 20, 1, pp. 1–6, 1973.
5. [5]
W. Specht, “Zur Theorie der Elementaren Mittel”, Math. Z. 74, 1960, pp. 91–98.