# Questions tagged [tree]

A tree is a special type of graph which only allows for a hierarchical set of edges similar to a tree . Mathematically it is actually an arborescence. Trees have a root node and children nodes. In formal terms it is described as an acyclic connected graph.

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### Capacitated Vehicle Routing- Help in understanding a proof

The paper "A Capacitated Vehicle Routing Problem on a Tree" (https://link.springer.com/content/pdf/10.1007/3-540-49381-6_42.pdf) by Shinya Hamaguchi1 and Naoki Katoh stated in the ...
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### Languages for specifying AST transformations with arbitrary arity

I'm developing a "computation assistant": users specify individual steps to take within an algebraic computation and the computer ensures they are applied correctly. I'm looking for ...
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### Complexity of finding a path visiting all leaves on a tree while respecting a distance bound

I am interested in the complexity of a specific variant of the Hamiltonian path problem where we want to visit all leaves of a tree while respecting a distance bound. Formally, given an (undirected, ...
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### Complexity of a problem related to Friedman's TREE(k) function?

Background Given two rooted, vertex-colored trees $T_1, T_2$, $T_1$ is color-preserving inf-embeddle in $T_2$, which we'll denote $T_1 \leq T_2$, if there is an injective $f \colon V(T_1) \to V(T_2)$ ...
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### Spanning Tree that Preserves the Number of Branch Vertices

Suppose a undirected connected graph $G$, denote the number of vertices in $G$ as $n$, number of branch vertices (i.e., vertices with degree $\geq 3$) as $n_{\geq 3}$. Suppose $n_{\geq 3}>\log(n)$. ...
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### O(n)-space, polylog-time subtree sums in incremental forests?

Consider a forest $G$ of $n$ vertices $v_1, \dots, v_n$ arranged left to right with edges from child to parent always going to the left, i.e. if the parent of vertex $v_i$ is $v_j$, then $j < i$. ...
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### Is the center of a BFS tree a good approximation of the graphs center?

Given a graph $G=(V,E)$, a center is a vertex $v\in V$ with minimal eccentricity (i.e., $v\in\text{argmin}_v\max_u d(u,v)$). Finding the center of the graph can easily be done using all-pairs-shortest-...
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### Directed tree decompositions on subtrees of DAGs

Given a DAG, is the arboreal decomposition of the DAG with the guarantee that given a node $x$, $v$ such that $x$ is reachable from $v$ are in the subtree of $x$? If not, is there a similar ...
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### Height of AVL tree with random elements

I know that for an AVL tree of N nodes, the depth of the tree is bounded by $$\log_2(N + 1) -1 \leq height \leq c \log_2(N + 2) + b$$ where $c,b$ are taken from the golden ratio linked to the worst ...
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### The number of rooted ordered trees of max-out degree $k$

An ordered tree (also known as ordinal tree and plane tree) is a rooted tree in which the children of each node are ordered. It is known that the number of the ordered tree with $n$ edges is the $n$'...
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### Distinguish Graph from Tree using Adjacency Matrix

Given an adjacency matrix, is there a way to determine if the graph will be a tree or a graph (whether or not there is a cycle). For example, given the adjacency matrix: ...
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### How to continue this algorithm? [closed]

I want to create an algorithm to fill a fixed-size big rectangle (W,H) with the maximum number of fixed-size smaller rectangles (w,h) (I can rotate the small rectangles 90º). I have thought about ...
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### Algorithm for computing unordered tree edit distance

I am trying to compute the edit distance between two dendrograms, one produced from hierarchical clustering, and the other manually constructed from some tree structure. In this setting, the rename ...
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### Place of tree-adjoining grammars in the hierarchy of tree grammars

As tree-adjoining grammars operate with trees, I suppose they can be considered as a kind of tree grammars. If this assumption is correct, I'm wondering: where should we place them in the tree grammar ...
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### Maximize the weight of MST + sum of vertex weights

I am considering a problem where the goal is to choose a subset of size $k$ of the vertices in a graph, such that the weight of their minimum spanning tree + the sum of their vertex weights is ...
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### Number of ordinal trees with n nodes, of depth d, with l leaves

What is the number of ordinal trees (aka rose trees) with $n$ nodes, of depth $d$, with $l$ leaves? I thought that it was a known results but I could not find it, and neither did the various ...
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### Finding a minimum tree which is isomorphic to a subtree of $T_1$ but not to a subtree of $T_2$

Consider the problem that receives two trees $T_1$, $T_2$, and asks to find a minimum size tree $T$ such that there exists a subtree of $T_1$ which is isomorphic to $T$, but there is no such ...
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### Inexact labelled binary tree matching

Does anyone recognise the following problems? Do they have names? Are they hard? If we were looking for an exact match (0 mismatches), these would be solvable in polynomial time (using e.g. standard ...
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### How to constrain a finite automaton (NFA and DFA) to a tree?

I have a finite automaton by the standard model Hopcroft & Ullman define: $$M = (Q, \Sigma, \delta, q_0, F)$$ Where $\delta$ is the transition function mapping $Q \times \Sigma \mapsto Q$, such ...
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### Exact formula for the number of spanning trees of a rectangle

This blog talks about generating "twisty little mazes" using a computer an enumerating them. The enumeration can be done using Wilson's algorithm to get the UST, but I don't remember the formula for ...
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### How does Camerini's algorithm for minimum-bottleneck-spanning-tree run in linear time?

I'm having a difficult time understanding Camerini's algorithm because there are very few clear explanations online. The goal is to find a minimum-bottleneck spanning tree in linear time. Camerini's ...
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### Suffix tree and searching for longest subword that appears two times and two occurencies are not overlapping

I'm learning basics of text algorithms, so my question might seem simple. Let's have word $S$, i want to find longest subword $x$ such that it appears in $S$ two times and those two occurencies are ...
If I must connect: $$2^k$$ terminals in a Steiner Tree choosen randomly and connect them with the cheapest component; "loss - contracting algorithm" is a good way? Or is an "Iterative Randomized ...
Could alternating tree automata be used for recognizing set (language) of arbitrary-arity trees? More specifically, as an example: let $\Sigma = \{a,b,c\}$ - labels for tree nodes. Trees from $T$ ...