Protection number in plane trees

Abstract

Abstract. The protection number of a plane tree is the minimal distance of the root to a leaf; this definition carries over to an arbitrary node in a plane tree by considering the maximal subtree having this node as a root. We study the the protection number of a uniformly chosen random tree of size n and also the protection number of a uniformly chosen node in a uniformly chosen random tree of size n. The method is to apply singularity analysis to appropriate generating functions. Additional results are provided as well.

Protection numbers in simply generated trees and Pólya trees

“…In 2017 Copenhaver [5] 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 infinity. These results were extended by Heuberger and Prodinger [18]. They showed the exact formulas for the first terms of the expectation, the variance and the probability of the respective protection numbers.…”

Protection numbers in simply generated trees and Pólya trees

“…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]. They showed the exact formulas for the rst terms of the expectation, the variance and the probability of the respective protection numbers.…”

“…We study both the root protection number as well as a random vertex protection number for the family of simply generated trees (introduced by Meir and Moon [23]) and their non-plane counterparts: unlabelled non-plane rooted trees, also called Pólya trees due to their rst extensive treatment by Pólya [26], examined further by Otter [24] including numerical results and the binary case. The present paper broadens the results from [17], but maintaining the emphasis on as concrete formulas as possible.…”

“…As in the previous section we calculate the mean via EY n = k≥1 P(Y n ≥ k). In order to do so we proceed analogously to Heuberger and Prodinger in [17] and dene S k (z) to be the generating function of the sequence (s n,k ) n≥0 of k-protected vertices summed over all trees of size n. As in [17] this generating function can be calculated by…”

“…converges rapidly. But (17) still bears a secret, because we do not have an explicit expression for S k (z) and we cannot solve the functional equation (15).…”