Enumeration of subtrees of trees

Abstract

Let $T$ be a weighted tree. The weight of a subtree $T_1$ of $T$ is defined as the product of weights of vertices and edges of $T_1$. We obtain a linear-time algorithm to count the sum of weights of subtrees of $T$. As applications, we characterize the tree with the diameter at least $d$, which has the maximum number of subtrees, and we characterize the tree with the maximum degree at least $\Delta$, which has the minimum number of subtrees.Comment: 20 pages, 11 figure

Subtrees of spiro and polyphenyl hexagonal chains

“…The number of leaf-containing subtrees of a binary tree provides a bound on the number of acceptable residue configurations, needed in the study of the complexity of a multiple parsimony alignment in phylogeny [14]. Much work has been done on this topic, including but not limited to, that of Szekely and Wang [41][42][43], Kirk [44], Czabarka [45], Vince [46], Zhang and Zhang [47,48], Yan and Yeh [49], Li [50], Jamison [51], Haslegrave [52], Andriantiana [53]. In particular, a linear-time algorithm to evaluate the sum of weights of subtrees of T through "generating functions" was provided in [49].…”

On the spectrum and number of convex sets in graphs

“…We showed, however, that for the class of cographs the g-convex sets can be enumerated in linear time. It was observed, using the results from [26], that the number g-convex sets of trees can also be enumerated in linear time.…”

“…In [26] it was shown that the number of subtrees of a tree can be determined in linear time. Since the subtrees of a tree are precisely the g-convex sets of a tree, this result shows that the number of g-convex sets of a tree can be determined in linear time.…”

On the distribution of subtree orders of a tree

“…This "subtree polynomial" was already considered by Jamison himself in [5]. More recently, Yan and Yeh [18] studied a weighted version, and Martin et al [9] considered a bivariate generalisation involving the number of leaves.…”