# Tight bounds on the path length of binary trees

## 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]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.CrossRefGoogle Scholar
- [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]D. E. Knuth, “The Art of Computer Programming”, vol. 3 “Sorting and Searching”, Addison-Wesley, reading, Mass., 1973.Google Scholar
- [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.CrossRefGoogle Scholar
- [5]W. Specht, “Zur Theorie der Elementaren Mittel”, Math. Z. 74, 1960, pp. 91–98.CrossRefGoogle Scholar