For questions about trees in graph theory, which are connected graphs with no cycles. Also can be used for questions about forests, which are graphs that are disjoint unions of trees.

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42 views

heavy path decomposition/heavy-light decomposition

I'm reading about the heavy path decomposition of trees and its application. I would like to know its time and space complexity. In addition, can the algorithm be implemented in distributed way? What ...
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14 views

Which cut-off for collapsing this tree?

I have a Newick tree that is built by comparing similarity (euclidean distance) of Position Weight Matrices (PWMs or PSSMs) of DNA regulatory motifs that are ~5-9 bp long sequences. An interactive ...
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29 views

prove splits compatible if and only if edge-split

"Prove that if $e_A$ and $e_B$ are distinct edges of a binary $X$-tree $T$ and $C=A\Delta B$(symmetric difference), then the splits $\sigma(A), \sigma(B)$ and $\sigma(C)$ are compatible if and only if ...
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13 views

What is a first-order dependency tree?

From the definition of Chow–Liu trees. Obviously, a tree is a graph with no loops; with first-order do they mean that it is a tree similar to those followed by a depth-first search?
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17 views

Representing trees in Set builder notation?

Is there a way to represent graphs and minimum spanning trees using set builder notation? e.g. I have a weighted graph of n nodes, all connected to each other in a mesh network manner. I am to ...
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1answer
33 views

Expected number of feed-forward/backward triangles in a random graph with internal nodes.

Suppose we have a graph with N* nodes (these are internal nodes. they all have at least one child). Every directed link in the network exists with probability p. What would be the expected number of: ...
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2answers
53 views

Finite trees and embedding in infinite regular trees.

Assume that you have a finite tree $T=(V,E)$, where $V$ and $E$ are the set of vertices and edges of $T$, respectively. Let $d_{max}$ be the maximum degree the some vertice(s) $v\in{V}$. Assume also ...
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39 views

How can I find the height of the Recursion Tree?

How do I determine the height of a Recursion tree? For example for the recursion $T(n) = 3T(\frac{2n}{3}) + O(1) $. Could you give me a hint?
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1answer
38 views

Proving that a sub-graph of a tree is a tree

The proof that P ::== any sub-graph, G* of the tree G, is also a tree, involves proof by contradiction. We can suppose that the sub-graph has a cycle --> the whole graph has a cycle --> the whole ...
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1answer
43 views

Definition of a leaf in a tree

Across two different texts, I have seen two different definitions of a leaf 1) a leaf is a node in a tree with degree 1 2) a leaf is a node in a tree with no children The problem that I see with ...
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1answer
357 views

Minimum number of nodes in balanced binary search tree

I'd like to know if anyone could help me verify a recursive formula for the minimum possible number of nodes a binary search tree would require to be balanced. So far, I know that the recursive ...
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68 views

Structural induction on internal nodes of a binary tree

I know my language is not super mathematic, but I want to make sure I have the logic down. Here is my proof for the number of internal nodes in a binary tree being equal to the floor_function(n/2), ...
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1answer
61 views

Binary Search Tree Traversals

Draw a BST when you insert, $O,V,E,R,F,L,C,W$ from left-to-right and determine the order of the nodes when using post-order traversal and pre-order traversal. My attempt at drawing the BST: ...
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34 views

Can an arbitrary network of nodes be effectively visualized as a circular “treemap”?

We all know that a treemap is effective for visualizing hierarchical tree data (i.e. where there are only 1 to many relationships like in a computer file system): But how difficult algorithmically ...
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1answer
48 views

Some questions about first-order logic (arising from a book by Raymond M Smullyan)

Recently, I got confused when reading a book about first order logic written by Raymond M smullyan. Chapter 1 page 9:When introducing the notion "Formation tree", smullyan define a formation tree for ...
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20 views

Can a search-tree be reversed in order to find more solutions?

consider a search-tree that we already know that if we start from the start-node (S), and go to nodes A1, B2, C1, then we get to the Goal-node. We have this solution, but we don't know the other nodes ...
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1answer
92 views

proof: center of a tree lies on the longest path

how can I make a proof of this property? I mean, given a weighted tree(with positive costs), how can I proof that the center of such a tree lies on the longest path?? I read to the first answer of ...
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1answer
59 views

Finding missing two edges in a MST in O(m) time

I need to write an algorithm in O(m) time to find the missing two edges of a minimum spanning tree. I am given a graph G(V,E) where m = |E| and n = |V| as an adjacency list, and T, a subset of G, with ...
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1answer
42 views

Recursive trees

Use the method of recursive tree to determine a good asymptotic upper bound (as tight as possible) for the following recurrence and prove your answer using induction (assuming that $T(n)$ is a ...
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32 views

Search L leaves smaller than node N

So in my binary $kd$-tree I have a node $N$. Now I search for the number of leafs $L$ "on the left" side of $N$ (this includes the left child branch of $N$ and all parents where the node is a right ...
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0answers
44 views

How to calculate branching factor of uniform tree

For a uniform tree of depth $d$ and if a particular problem has $N$ nodes then the $b*$ branching factor is $N + 1 = 1 + b* + (b*)^2 + ... + (b*)^d$. For a depth of 5 and N = 52 how is it that the ...
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2answers
50 views

Finding an equation for a growth formula

Given a tree that has three nodes each level I want to find the formula that predicts the number of all nodes with a given tree height. I fitted the data into Numbers with an exponential function ...
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102 views

max number of keys in a 2-3-4 tree

Let $M(L)$ be the largest number of keys (a $2$-node has $1$ key and two children, a $3$-node has $2$ keys and $3$ children, and a $4$-node has $3$ keys and $4$ children) in a $2-3-4$ tree that ...
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229 views

D ary tree node math

A d-ary tree is a rooted tree in which each node has at most d children (c) Suppose the tree has n nodes. What is the minimum the depth could possibly be, in terms of n and d? You can leave your ...
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1answer
60 views

Counting the number of trees on $[n]$

Let $T_{n}$ be the number of trees on $[n]$. Explain the identity below in terms of $T_{n}$ and prove it. $2(n-1)n^{n-2}=\sum_{k=1}^{n-1}\binom{n}{k}k^{k-1}(n-k)^{n-k-1}.$ So far I've got that ...
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16 views

Measuring values at nodes of two independent but now connected trees

I am not sure if this the right forum for this question and I hope I am providing enough details on what I want to accomplish. I have an application that has multiple trees - Tree 1 - is categories/ ...
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1answer
80 views

Unique solution for a given pre- and post-order of a rooted tree

Decide the picture of a rooted tree with pre-order $a,b,c,d,e,f,g,h$ and post-order $d,e,f,g,h,c,b,a$. Show that there always is a unique solution for a given pre- and post-order of a rooted tree. My ...
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1answer
57 views

Minimal Red-Black tree with depth 3

I'd like to ask what is minimal RBT with black depth 3. Is this following RBT ? Values are not important. And that tree can't have depth 2 or 1.
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21 views

Simplification of a dervived binary tree with n nodes [duplicate]

hi I need help with this problem how do simplify this equation and what are the steps and approaches to this problem
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0answers
298 views

What is the algorithm to sort 5 elements in 7 binary comparisons?

I'm tasked with finding the algo that sorts 5 elements in 7 binary comparisons. (The 7 is derived from ceilingFunction(log 5!), which our text states is the minimum number of comparisons required for ...
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1answer
42 views

Identify Type of Recursive Sequence?

I would love to learn techniques for solving the following, but I can't seem to identify this type of sequence: let $N > 0$ and let $k$ be an arbitrary positive integer between $0$ and $N-1$ ...
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106 views

Nilpotency of the adjacency matrix of a directed tree network

Say I have a directed network that is organized in a tree, with all connections going downstream (genealogically). By that I mean that there is one root node connected to $c_{00}$ child nodes, and ...
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93 views

Straight skeleton is a tree

Can anybody give me a hint on how to prove that the straight skeleton of every polygon is a tree. Here is the definition of the straight skeleton (taken from Wikipedia): The straight skeleton of a ...
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1answer
360 views

Is this formula for the number of nodes for a complete tree or a full and complete tree?

In a lecture it was said that "How many nodes are there in a complete k-ary tree with height h?" and this was the answer: $$ \sum^{h}_{i = 0}k^i $$ where h is the height and k is the max number of ...
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1answer
187 views

has deleting node in a binary search tree Displacement feature?

I am developing an academic project about graph and tree theory.I searched a lot but I didn't find a clear answer. In a part of project we want to delete some nodes from tree for example we want to ...
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1answer
606 views

M-ary tree problem

A full $m$-ary tree $T$ has 81 leaves and height 4 1) Give the upper and lower bounds for $m$ 2) What is $m$ if T is also balanced? [with $m^h=l$ for maximum leaf in a m-ary tree $m^4=81$ then m=3 ...
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1answer
359 views

Help in understanding search of Vantage-Point tree

This is my reference: http://stevehanov.ca/blog/index.php?id=130 A vantage-point tree is a way of organizing a set of points so that finding the n-nearest neighbors is as efficient as possible. It ...
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287 views

Depth-first spanning tree?

I am going to identify tree edges and back edges in an undirected graph. The graph consists of $5$ nodes, the edges between these nodes are as shown below: Suppose starting with $v_1$, after a ...
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49 views

An MST-like problem with vertex selection

Consider a planar pointset in a rectangle, where every point has a color (an integer label). We need to select one point of every color, so as to minimize the cost of a planar MST of selected points ...
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1answer
324 views

Explicit bijection between ordered trees with $n+1$ vertices and binary trees with $n+1$ leaves

What is an example of a direct bijection between ordered trees with $n+1$ vertices and binary trees with $n+1$ leaves?
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129 views

Terminology, mapping a tree to a tree

I have stumbled upon a problem, unfortunately I do not know the proper terminology to be used which hinders me in thinking about the problem and explaining the problem. I am not even sure this is the ...
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1answer
234 views

How to convert parentheses notation for trees into an actual tree drawing? [closed]

Trees are usually drawn as a set of objects connected by edges. But sometimes one sees a non-graphical, parentheses-based notation, like on the example below. What does the indentation mean in such ...
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1answer
108 views

The union of two connected graphs is connected [closed]

Let $G = (V,E)$ be a graph and let $H_1 = (V_1,E_1)$ and $H_2 = (V_2,E_2)$ be two connected subgraphs of $G$ that have at least one node in common. Prove that the graph $H = H_1\cup H_2 = (V_1\cup ...