Abstract
We determine the limit of the expected value and the variance of the protection number of the root in simply generated trees, in P?lya trees, and in unlabelled non-plane binary trees, when the number of vertices tends to infinity. Moreover, we compute expectation and variance of the protection number of a randomly chosen vertex in all those tree classes. We obtain exact formulas as sum representations, where the obtained sums are rapidly converging thus allowing an efficient numerical computation of high accuracy.
Highlights
The protection number of a tree is the length of the shortest path from the root to a leaf
We provide numerical values for several wellknown generated tree classes as well as for two non-plane classes studied in Sections 3 and 4
The asymptotic mean and variance for the protection number of a randomly chosen internal vertex in a random non-plane binary tree can be obtained in the same way as in the previous section for Pólya trees
Summary
The protection number of a tree is the length of the shortest path from the root to a leaf. In 2017 Copenhaver [4] found that in a random unlabelled plane tree the expected value of the protection number of the root and the expected value of the protection number of a random vertex approach 1.62297 and 0.727649, respectively, as the size of the tree tends to innity These results were extended by Heuberger and Prodinger [17]. The aim of this paper is to generalize the protection number results to a larger class of rooted trees We study both the root protection number as well as a random vertex protection number for the family of generated trees (introduced by Meir and Moon [23]) and their non-plane counterparts: unlabelled non-plane rooted trees, called Pólya trees due to their rst extensive treatment by Pólya [26], examined further by Otter [24] including numerical results and the binary case. We provide numerical values for several wellknown generated tree classes as well as for two non-plane classes studied in Sections 3 and 4
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