A Note on Weighted Rooted Trees

Let $T$ be a tree rooted at $r$. Two vertices of $T$ are related if one is a descendant of the other; otherwise, they are unrelated. Two subsets $A$ and $B$ of $V(T)$ are unrelated if, for any $a\in A$ and $b\in B$, $a$ and $b$ are unrelated. Let $\omega$ be a nonnegative weight function defined on $V(T)$ with $\sum_{v\in V(T)}\omega(v)=1$. In this note, we prove that either there is an $(r, u)$-path $P$ with $\sum_{v\in V(P)}\omega(v)\ge \frac13$ for some $u\in V(T)$, or there exist unrelated sets $A, B\subseteq V(T)$ such that $\sum_{a\in A }\omega(a)\ge \frac13$ and $\sum_{b\in B }\omega(b)\ge \frac13$. The bound $\frac13$ is tight. This answers a question posed in a very recent paper of Bonamy, Bousquet and Thomass\'e

Recognizing Treelike k-Dissimilarities

A k-dissimilarity D on a finite set X, |X| >= k, is a map from the set of size k subsets of X to the real numbers. Such maps naturally arise from edge-weighted trees T with leaf-set X: Given a subset Y of X of size k, D(Y) is defined to be the total length of the smallest subtree of T with leaf-set Y . In case k = 2, it is well-known that 2-dissimilarities arising in this way can be characterized by the so-called "4-point condition". However, in case k > 2 Pachter and Speyer recently posed the following question: Given an arbitrary k-dissimilarity, how do we test whether this map comes from a tree? In this paper, we provide an answer to this question, showing that for k >= 3 a k-dissimilarity on a set X arises from a tree if and only if its restriction to every 2k-element subset of X arises from some tree, and that 2k is the least possible subset size to ensure that this is the case. As a corollary, we show that there exists a polynomial-time algorithm to determine when a k-dissimilarity arises from a tree. We also give a 6-point condition for determining when a 3-dissimilarity arises from a tree, that is similar to the aforementioned 4-point condition.Comment: 18 pages, 4 figure

Enumeration of Weighted Plane Trees

In weighted trees, all edges are endowed with positive integral weight. We enumerate weighted bicolored plane trees according to their weight and number of edges.Comment: 6 pages, 4 figure

The combinatorics of the leading root of the partial theta function

Recently Alan Sokal studied the leading root $x_0(q)$ of the partial theta function $\Theta_0(x,q)=\sum\limits_{n=0}^\infty x^nq^{\binom n2}$, considered as a formal power series. He proved that all the coefficients of $$-x_0(q)=1+q+2q^2+4q^3+9q^4+...$$ are positive integers. I give here an explicit combinatorial interpretation of these coefficients. More precisely, I show that $-x_0(q)$ enumerates rooted trees that are enriched by certain polyominoes, weighted according to their total area.Comment: 15 pages, 7 figure