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Tree-decompositions and treewidth are of fundamental importance in structural and algorithmic graph theory. The "spread" of a tree-decomposition is the minimum integer $s$ such that every vertex lies in at most $s$ bags. A tree-decomposition is "domino" if it has spread 2, which is the smallest interesting value of spread. So that spread 1 becomes interesting, one can relax the definition of tree-decomposition to "tree-partition", which allows the endpoints of each edge to be in the same bag or adjacent bags, while demanding that each vertex appears in exactly one bag. Ding and Oporowski [1995] showed that every graph $G$ with treewidth $k$ and maximum degree $Δ$ has a tree-partition with width $O(kΔ)$. We prove the same result with an improved constant, and with the extra property that the underlying tree has maximum degree $O(Δ)$ and $O(|V(G)|/kΔ)$ vertices. This result implies (with an improved constant) the best known upper bound on the domino treewidth of $O(kΔ^2)$, due to Bodlaender [1999]. Moreover, solving an open problem of Bodlaender, we show this upper bound is best possible, by exhibiting graphs with domino treewidth $Ω(kΔ^2)$ for $k\geqslant 2$. On the other hand, allo

Recent advances in reinforcement learning (RL) have significantly enhanced the agentic capabilities of large language models (LLMs). In long-term and multi-turn agent tasks, existing approaches driven solely by outcome rewards often suffer from the problem of sparse supervision. To address the challenge, we propose Tree-based Group Relative Policy Optimization (Tree-GRPO), a grouped agent RL method based on tree search, where each tree node represents the complete agent interaction step. By sharing common prefixes, the tree search sampling increases the number of rollouts achievable within a fixed budget of tokens or tool calls. Moreover, we find that the tree-structured trajectory naturally allows the construction of step-wise process supervised signals even using only the outcome reward. Based on this, Tree-GRPO estimates the grouped relative advantages both on intra-tree and inter-tree levels. Through theoretical analysis, we demonstrate that the objective of intra-tree level group relative policy optimization is equivalent to that of step-level direct preference learning. Experiments across 11 datasets and 3 types of QA tasks demonstrate the superiority of the proposed tree-bas

Traditionally, AI development for two-player zero-sum games has relied on two primary techniques: decision trees and reinforcement learning (RL). A common approach involves using a fixed decision tree as one player's strategy while training an RL agent as the opponent to identify vulnerabilities in the decision tree, thereby improving its strategic strength iteratively. However, this process often requires significant human intervention to refine the decision tree after identifying its weaknesses, resulting in inefficiencies and hindering full automation of the strategy enhancement process. Fortunately, the advent of Large Language Models (LLMs) offers a transformative opportunity to automate the process. We propose RL-LLM-DT, an automatic decision tree generation method based on RL Evaluation and LLM Enhancement. Given an initial decision tree, the method involves two important iterative steps. Response Policy Search: RL is used to discover counter-strategies targeting the decision tree. Policy Improvement: LLMs analyze failure scenarios and generate improved decision tree code. In our method, RL focuses on finding the decision tree's flaws while LLM is prompted to generate an imp

Reinforcement learning (RL)-based post-training for large language models (LLMs) is computationally expensive, as it generates many rollout sequences that frequently share long token prefixes. Existing RL frameworks usually process these sequences independently during policy training, i.e., repeatedly recomputing identical prefixes in both the forward and backward passes of policy gradient computation, leading to substantial inefficiencies in computation resources and memory usage. Although prefix sharing naturally induces a tree structure over rollouts, packed tree-mask approaches scale poorly in RL settings. In this paper, we introduce AReaL-DTA, which efficiently exploits prefix sharing in RL training. AReaL-DTA employs a depth-first search (DFS)-based execution strategy that dynamically traverses the rollout prefix tree during both forward and backward computation, materializing only a single root-to-leaf path at a time. To further improve scalability, AReaL-DTA incorporates a load-balanced distributed batching mechanism that dynamically constructs and processes prefix trees across multiple GPUs. On $τ^2$-bench, AReaL-DTA improves training throughput by up to $8.31\times$ over

In 1998, it was shown that, if flavor SU(3) symmetry [SU(3)$_F$] is assumed in charmless $B \to PP$ decays ($P$ is a light pseudoscalar meson), some reduced matrix elements involving electroweak penguin (EWP) operators are related to those involving tree operators. Similarly, EWP diagrams are related to tree diagrams. These SU(3)$_F$ EWP-tree relations were recently used in global analyses of $B \to PP$ decays. They have also been used over the years in analyses of the $B \to πK$ puzzle, even though the $B \to πK$ amplitudes are related by isospin symmetry [SU(2)$_I$], and not the full SU(3)$_F$. In this paper, we show that, even if only SU(2)$_I$ is assumed, there are still EWP-tree relations. In $ΔS=0$ decays, these relations are similar to those of SU(3)$_F$, and can be used to take into account the EWP contributions in the extraction of the CP phase $α$ from $B \to ππ$ decays. In $ΔS=1$ decays, the SU(2)$_I$ EWP-tree relations are quite different from those of SU(3)$_F$; when these are used to analyze the $B \to πK$ puzzle, one now finds a 4-5$σ$ discrepancy with the Standard Model, much larger than what was previously found. We argue that, if one analyzes a set of hadronic $B$

We implemented a Soft Decision Tree (SDT) and a Short-term Memory Soft Decision Tree (SM-SDT) using PyTorch. The methods were extensively tested on simulated and clinical datasets. The SDT was visualized to demonstrate the potential for its explainability. SDT, SM-SDT, and XGBoost demonstrated similar area under the curve (AUC) values. These methods were better than Random Forest, Logistic Regression, and Decision Tree. The results on clinical datasets suggest that, aside from a decision tree, all tested classification methods yield comparable results. The code and datasets are available online on GitHub: https://github.com/KI-Research-Institute/Soft-Decision-Tree

Given an $\mathbb{N}$-weighted tree automaton, we give a decision procedure for exponential vs polynomial growth (with respect to the input size) in quadratic time, and an algorithm that computes the exact polynomial degree of growth in cubic time. As a special case, they apply to the growth of the ambiguity of a nondeterministic tree automaton, i.e. the number of distinct accepting runs over a given input. We deduce analogous decidability results (ignoring complexity) for the growth of the number of results of set queries in Monadic Second-Order logic (MSO) over ranked trees. In the case of polynomial growth of degree $k$, we also prove a reparameterization theorem for such queries: their results can be mapped to $k$-tuples of input nodes in a finite-to-one and MSO-definable fashion. We then apply these tools to study growth rates and subclass membership problems for tree-to-tree functions. Using new proof strategies, we recover and generalize known results concerning polyregular functions, total deterministic macro tree transducers, and partial nondeterministic top-down tree transducers. In particular, we give a procedure to decide polynomial size-to-height increase for both macr

Phylogenetic networks are a generalisation of phylogenetic trees that allow for more complex evolutionary histories that include hybridisation-like processes. It is of considerable interest whether a network can be considered `tree-like' or not, which lead to the introduction of \textit{tree-based} networks in the rooted, binary context. Tree-based networks are those networks which can be constructed by adding additional edges into a given phylogenetic tree, called the \textit{base tree}. Previous extensions have considered extending to the binary, unrooted case and the nonbinary, rooted case. We extend tree-based networks to the context of unrooted, nonbinary networks in three ways, depending on the types of additional edges that are permitted. A phylogenetic network in which every embedded tree is a base tree is termed a \textit{fully tree-based} network. We also extend this concept to unrooted, nonbinary phylogenetic networks and classify the resulting networks. We also derive some results on the colourability of tree-based networks, which can be useful to determine whether a network is tree-based.

Large-scale delineation of individual trees from remote sensing imagery is crucial to the advancement of ecological research, particularly as climate change and other environmental factors rapidly transform forest landscapes across the world. Current RGB tree segmentation methods rely on training specialized machine learning models with labeled tree datasets. While these learning-based approaches can outperform manual data collection when accurate, the existing models still depend on training data that's hard to scale. In this paper, we investigate the efficacy of using a state-of-the-art image segmentation model, Segment Anything Model 2 (SAM2), in a zero-shot manner for individual tree detection and segmentation. We evaluate a pretrained SAM2 model on two tasks in this domain: (1) zero-shot segmentation and (2) zero-shot transfer by using predictions from an existing tree detection model as prompts. Our results suggest that SAM2 not only has impressive generalization capabilities, but also can form a natural synergy with specialized methods trained on in-domain labeled data. We find that applying large pretrained models to problems in remote sensing is a promising avenue for futu

We study Stanley's chromatic symmetric function (CSF) for trees when expressed in the star basis. We use the deletion-near-contraction (DNC) algorithm to compute coefficients that occur in the CSF in the star basis. In particular, one of our main results determines the smallest partition in lexicographic order that occurs as an indexing partition in the CSF, and we also give a formula for its coefficient. In addition to describing properties of trees encoded in the coefficients of the star basis, we give an algorithm for reconstructing trees of diameter less than six.

The goal of these lectures is to review some mathematical aspects of random tree models used in evolutionary biology to model gene trees or species trees. We start with stochastic models of tree shapes (finite trees without edge lengths), culminating in the $β$-family of Aldous' branching models. We next introduce real trees (trees as metric spaces) and show how to study them through their contour, provided they are properly measured and ordered. We then focus on the reduced tree, or coalescent tree, which is the tree spanned by individuals/species alive at the same fixed time. We show how reduced trees, like any compact ultrametric space, can be represented in a simple way via the so-called comb metric. Beautiful examples of random combs include the Kingman coalescent and coalescent point processes. We end up displaying some recent biological applications of coalescent point processes to the inference of species diversification, to conservation biology and to epidemiology.

World models have emerged as a pivotal component in robot manipulation planning, enabling agents to predict future environmental states and reason about the consequences of actions before execution. While video-generation models are increasingly adopted, they often lack rigorous physical grounding, leading to hallucinations and a failure to maintain consistency in long-horizon physical constraints. To address these limitations, we propose Embodied Tree of Thoughts (EToT), a novel Real2Sim2Real planning framework that leverages a physics-based interactive digital twin as an embodied world model. EToT formulates manipulation planning as a tree search expanded through two synergistic mechanisms: (1) Priori Branching, which generates diverse candidate execution paths based on semantic and spatial analysis; and (2) Reflective Branching, which utilizes VLMs to diagnose execution failures within the simulator and iteratively refine the planning tree with corrective actions. By grounding high-level reasoning in a physics simulator, our framework ensures that generated plans adhere to rigid-body dynamics and collision constraints. We validate EToT on a suite of short- and long-horizon manip

We propose a general data structure CORoBTS for storing B-tree-like search trees dynamically in a cache-oblivious way combining the van Emde Boas memory layout with packed memory array. In the use of the vEB layout mostly search complexity was considered, so far. We show the complexity of depth-first search of a subtree and contiguous memory area and provide better insight into the relationship between positions of vertices in tree and in memory. We describe how to build an arbitrary tree in vEB layout if we can simulate its depth-first search. Similarly, we examine batch updates of packed memory array. In CORoBTS, the stored search tree has to satisfy that all leaves are at the same depth and vertices have arity between the chosen constants $a$ and $b$. The data structure allows searching with an optimal I/O complexity $\mathcal{O}(\log_B{N})$ and is stored in linear space. It provides operations for inserting and removing a subtree; both have an amortized I/O complexity $\mathcal{O}(S\cdot(\log^2 N)/B + \log_B N\cdot\log\log S + 1)$ and amortized time complexity $\mathcal{O}(S\cdot\log^2 N)$, where $S$ is the size of the subtree and $N$ the size of the whole stored tree. Rebuildi

Generating accurate digital tree models from scanned environments is invaluable for forestry, agriculture, and other outdoor industries in tasks such as identifying biomass, fall hazards and traversability, as well as digital applications such as animation and gaming. Existing methods for tree reconstruction rely on feature identification (trunk, crown, etc) to heuristically segment a forest into individual trees and generate a branch structure graph, limiting their application to sparse trees and uniform forests. However, the natural world is a messy place in which trees present with significant heterogeneity and are frequently encroached upon by the surrounding environment. We present a general method for extracting the branch structure of trees from point cloud data, which estimates the structure of trees by adapting the methods of structural topology optimisation to find the optimal material distribution to support wind-loading. We present the results of this optimisation over a wide variety of scans, and discuss the benefits and drawbacks of this novel approach to tree structure reconstruction. Despite the high variability of datasets containing trees, and the high rate of occ

Let $G$ be a graph and $S\subseteq V(G)$ with $|S|\geq 2$. Then the trees $T_1, T_2, \cdots, T_\ell$ in $G$ are \emph{internally disjoint Steiner trees} connecting $S$ (or $S$-Steiner trees) if $E(T_i) \cap E(T_j )=\emptyset$ and $V(T_i)\cap V(T_j)=S$ for every pair of distinct integers $i,j$, $1 \leq i, j \leq \ell$. Similarly, if we only have the condition $E(T_i) \cap E(T_j )=\emptyset$ but without the condition $V(T_i)\cap V(T_j)=S$, then they are \emph{edge-disjoint Steiner trees}. The \emph{generalized $k$-connectivity}, denoted by $κ_k(G)$, of a graph $G$, is defined as $κ_k(G)=\min\{κ_G(S)|S \subseteq V(G) \ \textrm{and} \ |S|=k \}$, where $κ_G(S)$ is the maximum number of internally disjoint $S$-Steiner trees. The \emph{generalized local edge-connectivity} $λ_{G}(S)$ is the maximum number of edge-disjoint Steiner trees connecting $S$ in $G$. The {\it generalized $k$-edge-connectivity} $λ_k(G)$ of $G$ is defined as $λ_k(G)=\min\{λ_{G}(S)\,|\,S\subseteq V(G) \ and \ |S|=k\}$. These measures are generalizations of the concepts of connectivity and edge-connectivity, and they and can be used as measures of vulnerability of networks. It is, in general, difficult to compute these

Given a set $P$ of $n$ points in $\mathbb{R}^2$ and an input line $γ$ in $\mathbb{R}^2$, we present an algorithm that runs in optimal $Θ(n\log n)$ time and $Θ(n)$ space to solve a restricted version of the $1$-Steiner tree problem. Our algorithm returns a minimum-weight tree interconnecting $P$ using at most one Steiner point $s \in γ$, where edges are weighted by the Euclidean distance between their endpoints. We then extend the result to $j$ input lines. Following this, we show how the algorithm of Brazil et al. ("Generalised k-Steiner Tree Problems in Normed Planes", arXiv:1111.1464) that solves the $k$-Steiner tree problem in $\mathbb{R}^2$ in $O(n^{2k})$ time can be adapted to our setting. For $k>1$, restricting the (at most) $k$ Steiner points to lie on an input line, the runtime becomes $O(n^{k})$. Next we show how the results of Brazil et al. ("Generalised k-Steiner Tree Problems in Normed Planes", arXiv:1111.1464) allow us to maintain the same time and space bounds while extending to some non-Euclidean norms and different tree cost functions. Lastly, we extend the result to $j$ input curves.

In the first paper (part I) of this series of two, we introduce four novel definitions of the ODT problems: three for size-constrained trees and one for depth-constrained trees. These definitions are stated unambiguously through executable recursive programs, satisfying all criteria we propose for a formal specification. In this sense, they resemble the "standard form" used in the study of general-purpose solvers. Grounded in algebraic programming theory-a relational formalism for deriving correct-by-construction algorithms from specifications-we can not only establish the existence or nonexistence of dynamic programming solutions but also derive them constructively whenever they exist. Consequently, the four generic problem definitions yield four novel optimal algorithms for ODT problems with arbitrary splitting rules that satisfy the axioms and objective functions of a given form. These algorithms encompass the known depth-constrained, axis-parallel ODT algorithm as the special case, while providing a unified, efficient, and elegant solution for the general ODT problem. In Part II, we present the first optimal hypersurface decision tree algorithm and provide comprehensive experim

With the exponential growth of research facilitated by modern technology and improved accessibility, scientific discoveries have become increasingly fragmented within and across fields. This makes it challenging to assess the significance, novelty, incremental findings, and equivalent ideas between related works, particularly those from different research communities. Large language models (LLMs) have recently demonstrated strong quantitative and qualitative reasoning abilities, and multi-agent LLM debates have shown promise in handling complex reasoning tasks by exploring diverse perspectives and reasoning paths. Inspired by this, we introduce Tree-of-Debate (ToD), a framework which converts scientific papers into LLM personas that debate their respective novelties. To emphasize structured, critical reasoning rather than focusing solely on outcomes, ToD dynamically constructs a debate tree, enabling fine-grained analysis of independent novelty arguments within scholarly articles. Through experiments on scientific literature across various domains, evaluated by expert researchers, we demonstrate that ToD generates informative arguments, effectively contrasts papers, and supports re

A tree view or tree navigator is used to display hierarchical data organized in the form of a tree. In a tree structure there are parent and child nodes. The child nodes may further have descendants to n levels. There are many methods to make the navigation easy. Some of these are expanding and collapsing branches, splitting the tree, displaying a parent node in a separate tree, zooming branches, scrolling in various directions etc. It is still a difficult exercise to handle large trees efficiently. The effort still continues to manage large number of nodes with faster speed, greater control, user friendliness and aesthetics. This article illustrates five inventions on tree navigators selected from US patent database. Each of them tries to solve various problems relating to the tree navigator in different ways. Each invention is also analyzed from a TRIZ perspective.