# Questions tagged [trees]

Questions about a special kind of graphs, namely connected and cycle-free ones.

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31 views

### Determine minimum and maximum number of leaves on a complete tree

I want to determine the minimum and maximum number of leaves of a complete tree(not necessarily a binary tree) of height $h$. I already know how to find minimum($h+1$) and maximum($2^{h+1}-1$) number ...
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### What are some generalizations of the tree diameter?

I am specifically interested in extensions and generalizations of the diameter concept on graphs. I can't find anything relevant in google scholar. I have a certain result about tree diameters that ...
41 views

### Count total number of k length paths in a tree

This is a question from a competitive programming competition. Given a tree with n nodes and a number k, find the total number of paths of length k in that tree. I know for a fact that a solution can ...
108 views

### Thought process to solve tree based Dynamic Programming problems

I am having a very hard time understanding tree based DP problems. I am fairly comfortable with array based DP problems but I cannot come up with the correct thought process for tree based problems ...
28 views

### Prove that there is a sequence of k minimum spaning trees between two distinct minimum spanning trees that each one is different in only 1 edge [duplicate]

I'm pracitcing exams towards finals, Given an undirected graph $G(V,E)$ , we denote 2 MST $T,T'$ neighbours if by deleting one edge from $T$ and add another one we get $T'$. Prove : for every 2 ...
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### Traversing/comparing tree structures with only parent references

Consider a tree structure with nodes containing references to only their parent. So, the root node's parent will be null. This tree represents a class hierarchy. The goal is to search such a tree for ...
60 views

### What's the number of leaves in AVL tree

How can I prove that the number of leaves in a balanced BST is $\Omega (N)$ where $N$ is the number of nodes in the tree? I tried somehow to prove that an AVL/Fibonacci tree should have $\Omega (N)$ ...
32 views

### Why multiple rotations might be needed after deletion in an AVL tree if after insertion there can be at most one needed?

I understand that after deletion you have to retrace to update ancestors and after insertion you do the similar however at most one rotation will be performed. The question is why is there the ...
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### Does this kind of “conversation tree” has a specific name?

Does a Tree of infinite width and depth where each node represents a syntactically valid response to (a node of) a previous response to which it's connected via an edge has a specific name? (It seems ...
34 views

### How to delete a node from BST tree with 2 chidren?

I googled, read several tutorials and watched several BST node deletion algorithm explanations before posting this question. For some reason, I cannot find a complete explanation of BST node deletion ...
44 views

### Joining line segments to make tree

Given a set of disjoint line segments in the plane, prove (or disprove) that we can always join the line segments to make a tree where the vertices of the tree are the endpoints of the segments and ...
53 views

### Max nodes whose value exceeds all neighbors

A node is valid if its value is greater than all of its incident edges. Task is to maximize the number of valid nodes. Given $n$ values for nodes and $n-1$ values for edges, how do I assign these ...
32 views

### Prove G have a single MSP

We have undirected connective, weighted graph $G = (V,E)$. we also know that for every $e,e'$ in $E$, $w(e)≠w(e')$. Prove that $G$ has a single MSP. Ideas?
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### Enumerate all paths of length 3 in a given tree T

Kind help with an algorithm or any refrence to enumerate all paths of length 3 in a given tree T in the shortest possible time.
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### Counting nodes within K distance from set of given nodes in a tree [duplicate]

I was going through this article https://www.geeksforgeeks.org/count-nodes-within-k-distance-from-all-nodes-in-a-set/ The question says: Given an undirected tree with some marked nodes and a positive ...
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### Scapegoat Trees: Why are they only loosely a-height-balanced?

From Wikipedia: Even a degenerate tree (linked list) satisfies this condition if α=1, whereas an α=0.5 would only match almost complete binary trees. A binary search tree that is α-weight-...
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### Red-Black tree with index

I want to create a Red-Black Tree, with 2 values, (index, value) and I want to insert into the RB_tree based on the index. So if I have the function: $\text{insert}(root, value, index)$ it will ...
20 views

### Finding most likely tree over a semilattice

If I am not mistaken, then a semilattice defines a finite set of trees, for example spanning trees. Now assume that each semilattice edge is annotated with a transition probability. In addition, let'...
30 views

### Name for Turning DAG into redundant tree

I am looking for a term: How is the tree called that you can obtain from a DAG by going top-down and appending all visited nodes to a tree, thereby copying nodes from the DAG into multiple occurences ...
124 views

### Solving $T(n) = T(n/2) + T (n/3) + n$ with recurrence tree

I am trying to solve the following recurrence relation: $$T(n) = T(n/2) + T (n/3) + n$$ $$T(1) = Θ(1)$$ I guess that the time complexity is $T(n)=Θ(n)$ since $\frac{n}{2} + \frac{n}{3} < n$ I ...
49 views

### Which Tree traversal String is unique?

Assume we have a tree and we want to serialize it. Example: ...
105 views

### Generate random labeled tree with constrained edge lengths

Let $T$ be a labeled tree with vertices $V = \{1, \dots, n\}$ and edges $E$. Define the length of an edge $e = \{ u, v \}, u \in V, v \in V$ to be $l(e) = |u - v|$, i.e. the distance between the nodes ...
23 views

### how to construct a tree using root tree's code?

The Question is if we are given a binary code how construct a tree using that code? There was a question as follows: By root tree's 000101001111 code, reconstruct that tree. and the complete answer: ...
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### Data structure for storing strings

I'm designing a tree data structure to store strings in. One classic solution is prefix tree, but I am looking for a solution that the time to check if the string is in the storage is O(logm*logn) ...
126 views

### How to answer multiple queries for a tree?

I encountered an interesting problem based on tree-data-structure. We are given a tree which has N nodes, with 1≤N≤105. Time starts from second 1 and it continues for q seconds. At each ...
104 views

### How to query the tree?

I encountered an interesting problem based on tree-data-structure. We are given a tree which has N nodes, with 1≤N≤105. Time starts from second 1 and it continues for q seconds. At each ...
54 views

### Recursive algorithm for finding a common ancestor between two nodes in a tree, if it exists?

Here's the start and the vernacular I'm using. ...
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### Leaf nodes of B+ Tree

I have a b+ tree and i want to find the record associated with a specific key Ki. So i run the b+ tree search algorithm. If a certain node in the search path is a leaf and K=Ki, then the record exists ...
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### Average number of full nodes in rooted m-ary tree

I am looking for a formula to express the average number of full nodes (i.e. nodes having exactly $m$ children) in a $m$-ary tree having $n$ nodes, i.e.,  \mu_{n}^{(m)} = \frac{\# \text{full nodes ...
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### Traverse tree collecting nodes combinations

Summary: I have a tree, containing car parts, from which I need to build all valid combinations of these parts. Parts can be required - every build must contain them. Parts can be optional. Parts ...
55 views

### A* algorithm example explained

I am trying to understand A star algorithm. I am aware that it follows the sum of the current cost and heuristic, therefore f(n) = c(n) + h(n) in order to expand a ...
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### How to find running time complexity of divide and conquer method without Master Theorem

I understand that Master Theorem can be used to solve divide-and-conquer run times if they're in the form of $T(n) = aT(\frac{n}{b}) + n^clog^k(n)$ The reason behind it has to do with drawing a tree ...
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### How to find all the edges shared by all diametral paths of a tree?

A diametral path in a graph is a shortest path whose length is equal to the diameter of the graph. Now, given a tree with $n$ nodes, I would like to find the set of edges (possibly empty) which are ...
48 views

### How to solve following tree problem?

Source of the problem is https://codeforces.com/contest/1152/problem/D . I think i understood the problem , but if possible please explain the problem in simple way and the solution also .I tried very ...
26 views

### RB trees from any balanced BST?

Given any perfectly balanced binary search tree, is it always possible to assign a coloring to the nodes so that it becomes a Red-Black tree? If so, how do you prove this, and if false, what would be ...
48 views

### Merge nodes in a graph to form a tree

I have an undirected graph which may have 2 edges that connect the same pair of nodes. A group of nodes that don't need any bridges to go from a node to another should be merged into one node in a ...
127 views

### Maximum space consumption of stack and queue for DFS and BFS

I'm trying to determine the maximum memory consumption of the "pending nodes" data structure (stack/queue) for both travelings: BFS and (preorder) DFS. Since BFS and DFS while traveling graphs have ...
29 views

### Greedy algorithm for feedback vertex set / greedy algorithms vs local ratio in general

A greedy algorithm for finding a minimum feedback vertex set is to pick and remove a vertex with minimum $w(v)/\delta_H(v)$, where $H$ is the current graph, until there are no more cycles left. (...
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### How to do a reverse topological sort using depth first search?

I'm doing a replacement for the venerable make utility that will support, among other things, automatic cleaning. The utility figures out automatically what files ...
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### Efficiently calculate values based on subsets of a list

Let's say I have a list of items, each of which has some value Eg. L1 = [A, B, C, ...] And another list of items L2: [1, 2, 3, ...]. Each of the items in L2 needs to calculate the max value of a ...
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### Disjoint Set Connected Components With Weighted Graph

I have been trying to solve this HackerRank problem (link). The basic premise of this problem is that there is a tree with undirected, but weighted, edges. The cost of a path in this tree is taken ...
161 views

### Min Fibonacci Heap - increase key

I have been trying to implementing heap data structures for use in my research work. As part of that, I am trying to implement increase-key operations for min-heaps....
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### Minimum Weight Binary Spanning Tree

Let $G=(V,E)$ be a simple graph with weights $w_{ij}$ (can be assumed to be positive). Is it possible to find the minimum (or maximum) weight, rooted spanning tree that is binary? That means every ...
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### Directed Trees: Finding all the edges and vertices in a specific direction

I am an electrical engineer without experience in graph theory. However, I have a problem which I believe can be solved by graph theory. We have a directed tree, such as the one below. We want to find ...
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### Minimum number of moves to reach a grid point by modified knight in variant chessboard

I apologize if this is not the right board to post this question but I'm cross-posting from the mathematics board. I am dealing with a computational question that extends the question posed in https://...
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### Depth of an R-tree, given $m$, $M$ and number of elements

Simply: what is the theoretical maximum, minimum or expected depth of an R-tree given $m$ minimum $M$ maximum elements in a node, with $N$ amount of nodes?
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### How to merge a lot of trees into one single graph?

I have a few different trees, which resemble what the AST that compilers often deal with. For example: tree 1 ( (a, b), (c, d) ) Imagine that each tree split represents the function "add", then ...
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### Is Edmonds' Matroid partitioning algorithm optimal w.r.t lexicographical order?

We all know that, given a matroid $(E, \mathcal{I})$, Edmonds' Matroid partitioning algorithm will result in a tuple of $E$-covering, pairwise-disjoint independent sets $(I_1, ..., I_k)$ with optimal (...