On the shape of the fringe of various types of random trees

Abstract

Abstract. We analyze a fringe tree parameter w in a variety of settings, utilizing a variety of methods from the analysis of algorithms and data structures.Given a tree t and one of its leaves a, the w(t, a) parameter denotes the number of internal nodes in the subtree rooted at a's father. The closely-related w(t, a) parameter denotes the number of leaves, excluding a, in the subtree rooted at a's father. We define the cumulative w parameter as W(t) = a w(t, a), i.e., as the sum of w(t, a) over all leaves a of t. The w parameter not only plays an important rôle in the analysis of the Lempel-Ziv '77 data compression algorithm, but it is captivating from a combinatorial viewpoint too.In this report, we determine the asymptotic behavior of the w and W parameters on a variety of types of trees. In particular, we analyze simply generated trees, recursive trees, binary search trees, digital search trees, tries and Patricia tries.The final section of this report briefly summarizes and improves the previously known results about the w parameter's behavior on tries and suffix trees, originally published in one author's thesis (see [War05]This survey of new results about the w parameter is very instructive since a variety of different combinatorial methods are used in tandem to carry out the analysis.

Fringe trees, Crump–Mode–Jagers branching processes and $m$-ary search trees

“…Since 2) we find by Theorem A.5 (with χ = ρ = 1, α = 2) X − 1 ∼ HG(1, 1; 4). In other words, using (B.3)-(B.5), (12.5) This limit distribution was found by Drmota, Gittenberger, Panholzer, Prodinger and Ward [47] (in a slightly different setting, keeping track of the position of the nodes).…”

“…Since This limit distribution was found by Drmota, Gittenberger, Panholzer, Prodinger and Ward [47] (in a slightly different setting, keeping track of the position of the nodes).…”

“…Of course, the fringe tree T v rooted at v is trivial, so the interest is in the extended fringe and in properties of the type in Theorem 5.25. For example, Drmota, Gittenberger, Panholzer, Prodinger and Ward [47] study the number of internal nodes (and the number of leaves) in the subtree rooted at the father of a randomly chosen leaf, for a variety of different types of random trees.…”

“…This describes, at least in principle, any properties of T 1 . For example, we may as in [47] count leaves and non-leaves separately in T 1 . It is easy to see by induction that if k 2, then P(T k has i leaves) = P(T k has i non-leaves) =…”

An analytic approach to the asymptotic variance of trie statistics and related structures

“…We define the peripheral path length of a tree as the sum of the fringe-sizes of all leaf-nodes, where the fringe-size of a leaf is defined to be the number of external nodes of the subtree rooted at its parent-node. This parameter was investigated in [16] where it was called the wparameter. It was also studied in phylogenetics in the context of sum of all minimal clade sizes (see [4]).…”