Abstract
We study protected nodes in various classes of random rooted trees by putting them in the general context of fringe subtrees introduced by Aldous (1991). Several types of random trees are considered: simply generated trees (or conditioned Galton-Watson trees), which includes several cases treated separately by other authors, binary search trees and random recursive trees. This gives unified and simple proofs of several earlier results, as well as new results.
Highlights
We study protected nodes in various classes of random rooted trees by putting them in the general context of fringe subtrees introduced by Aldous (1991)
The purpose of the present paper is to extend and sharpen some of these results by putting them in the general context of fringe subtrees introduced by Aldous [1]
The first class of random trees that we consider in this paper are the generated random trees; these are defined using a weight sequence∞ k=0 which we regard as fixed, see Section 2 for the definition and the connection to conditioned Galton– Watson trees. It is well-known that suitable choices of∞ k=0 yield several important classes of random trees, see e.g. Aldous [2], Devroye [6], Drmota [9], Janson [14] and
Summary
Several recent papers study protected nodes in various classes of random rooted trees, where a node is said to be protected if it is not a leaf and, none of its children is a leaf. (Equivalently, a node is protected if and only if the distance to any descendant that is a leaf is at least 2; for generalizations, see Section 5.) See Cheon and Shapiro [5] (uniformly random ordered trees, Motzkin trees, full binary trees, binary trees, full ternary trees), Mansour [18] (k-ary trees), Du and Prodinger [10] (digital search trees), Mahmoud and Ward [16] (binary search trees), Mahmoud and Ward [17] (random recursive trees), Bóna [4] (binary search trees). The first class of random trees that we consider in this paper are the generated random trees; these are defined using a weight sequence (wk)∞ k=0 which we regard as fixed, see Section 2 for the definition and the connection to conditioned Galton– Watson trees. It is well-known that suitable choices of (wk)∞ k=0 yield several important classes of random trees, see e.g. Aldous [2], Devroye [6], Drmota [9], Janson [14] and Section 4. As far as we know, the fringe subtrees of these random trees have not been studied in general; this will be dealt with elsewhere
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