STACS 1984: STACS 84 pp 109-120

# On a general weight of trees

Contibuted Papers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 166)

## Abstract

We define a general weight of the nodes of a given tree T; it depends on the structure of the subtrees of a node, on the number of interior and exterior nodes of these subtrees and on three weight functions defined on the degrees of the nodes appearing in T. Choosing particular weight functions, the weight of the root of the tree is equal to its internal path length, to its external path length, to its internal degree path length, to its external degree path length, to its number of nodes of some degree'r, etc.

For a simply generated family of rooted planar trees (e.g. all trees defined by restrictions on the set of allowed node degrees), we shall derive a general approach to the computation of the average weight of a tree T ε with n nodes and m leaves for arbitrary weight functions, on the assumption that all these trees are equally likely. This general result implies exact and asymptotic formulas for the average weight of a tree T ε with n nodes for arbitrary weight functions satisfying particular conditions. Furthermore, this approach enables us to derive explicit and asymptotic expressions for the different types of average path lengths of a tree T ε with n nodes and of all ordered trees with n nodes and m leaves.

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### References

1. [1]
Bender, E.A.: Central and Local Limit Theorems Applied to Asymptotic Enumeration, J. Comb. Theory (Ser. B) 15, 95–111 (1973)Google Scholar
2. [2]
Bender, E.A.: Asymptotic Methods in Enumeration, SIAM Rev. 16(4), 485–515 (1974)Google Scholar
3. [3]
Flajolet, Ph., Odlyzko, A.: The Average Height of Binary Trees and Other Simple Trees, JCSS 25(2), 171–213 (1982)Google Scholar
4. [4]
Kemp, R.: On the Average Oscillation of a Stack, Combinatorica 2(2), 157–176 (1982)Google Scholar
5. [5]
Kemp, R.: The Average Height of Planted Plane Trees with M Leaves, J. Comb. Theory (Ser. B) 34(2), 191–208 (1983)Google Scholar
6. [6]
Knuth, D.E.: The Art of Computer Programming, vol. 1, 2nd ed. Addison Wesley, Reading, Mass. 1973Google Scholar
7. [7]
8. [8]
Meir, A., Moon, J.W.: On the Altitude of Nodes in Random Trees, Can. Math. 30, 997–1015 (1978).Google Scholar