Extremal Values of Ratios: Distance Problems vs. Subtree Problems in Trees

The authors discovered a dual behaviour of two tree indices, the Wiener index and the number of subtrees, for a number of extremal problems [Discrete Appl. Math. 155 (3) 2006, 374-385; Adv. Appl. Math. 34 (2005), 138-155]. Barefoot, Entringer and Székely [Discrete Appl. Math. 80 (1997), 37-56] determined extremal values of σT(w)/σT(u), σT(w)/σT(v), σ(T)/σT(v), and σ(T)/σT(w), where T is a tree on n vertices, v is in the centroid of the tree T, and u,w are leaves in T. In this paper we test how far the negative correlation between distances and subtrees go if we look for the extremal values of FT(w)/FT(u), FT(w)/FT(v), F(T)/FT(v), and F(T)/FT(w), where T is a tree on n vertices, v is in the subtree core of the tree T, and u,w are leaves in T-the complete analogue of [Discrete Appl. Math. 80 (1997), 37-56], changing distances to the number of subtrees. We include a number of open problems, shifting the interest towards the number of subtrees in graphs

Greedy Trees, Subtrees and Antichains

Greedy trees are constructed from a given degree sequence by a simple greedy algorithm that assigns the highest degree to the root, the second-, third-, ... highest degrees to the root\u27s neighbors, and so on. They have been shown to maximize or minimize a number of different graph invariants among trees with a given degree sequence. In particular, the total number of subtrees of a tree is maximized by the greedy tree. In this work, we show that in fact a much stronger statement holds true: greedy trees maximize the number of subtrees of any given order. This parallels recent results on distance-based graph invariants. We obtain a number of corollaries from this fact and also prove analogous results for related invariants, most notably the number of antichains of given cardinality in a rooted tree