# Questions tagged [binary-trees]

a tree in which each node has no more than two children

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613 views

### Chained operations on sequences with two operators

Given a binary expresion tree, with addition and multiplication operations, how can we optimize it's evaluation? Can we learn from matrix chain multiplication? A generalization of matrix chain ...
150 views

### Correctness of a greedy Algorithm on Knockout Tournaments

You are given a function $\operatorname{rk}:\{1\dots 2^k\}\rightarrow \mathbb{N^+}$ representing the ranks of the players $1\dots2^k$ in a participating in a tournament. The tournament evolves in a ...
250 views

### Find if a sequence of node is a result of a binary tree preorder traversal

Assume we have a sequence of 0's and 1's: $n_0, n_1, ..., n_N$, in which 0 stands for a leaf node, and 1 stands for an uncertain node (it may or may not be a leaf node). How to check if this sequence ...
222 views

### Average redundancy in Huffman or Hu-Tucker codes on random symbol probabilities

Huffman and Hu-Tucker codes are well-known compression schemes, which both come close to the entropy lower bound. It is known that if $L_1$ and $L_2$ are the lengths of a Huffman resp. Hu-Tucker code, ...
290 views

### Efficient deletion in Fenwick tree

Suppose there is an array of nonzero integer values A[n], which has a Fenwick tree representation F[n]. The most simplistic way ...
75 views

### Number of binary trees of size $n$ such that all subtrees of same size are equal?

In the following, I consider rooted, unlabelled, ordered binary trees, where each node has exactly $0$ or $2$ children (I will simply call them binary trees). A binary tree $t'$ is a subtree of a ...
580 views

### Optimal Binary Search Trees Knuth

Knuth, Donald E. (1971), "Optimum binary search trees", Acta Informatica 1 (1): 14–25,doi:10.1007/BF00264289 Please have a look at this paper, specifically page 18 in which he tries to prove his ...
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### Node depth in randomly built binary search tree

It can be proved that randomly built binary search trees of size $n$ are of depth $O(\log n)$ and it is clear that level $k$ has at most $2^{k}$ nodes (root's level is 0). I have an algorithm that ...
205 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 $\Omega (N)$ ...
37 views

### Proving an upper-bound on the cost of all calls to rebuild() during a sequence of m operations in a scapegoat tree

I'm reading through Open Data Structures by Pat Morin and am currently looking at scapegoat trees. The book is free and can be downloaded from here: http://opendatastructures.org/ods-java.pdf On page ...
146 views

### Total number of red black tree arrangements

Consider a list of n integers. And assume we take each integer from this list sequentially and add it to a red-black tree. Then for the n! permutations of the list how many red-black tree ...
275 views

### What is the intuition behind balancing in AVL trees?

I am not sure that my question is clear from the first sight. But I will try to explain what I mean. For now, I am learning balancing the trees on the example of AVL trees. We know that to balance ...
535 views

### Range bit inversions and range set bit queries with a binary indexed tree

I am trying to solve this problem using a binary indexed tree. The problem can be summarized as follows: You are given a series of commands that operate on an array initially all zeroes. 1) ...
828 views

### what is the name for the space between the leaves of a tree

I am trying to write a data-type not for a tree, but for the spaces in between the leaves of thee tree. In number theory (a part of math) this is known as a topograph does it have a name in CS?
24 views

### Nested dissection vs kd-tree

Could you explain, please, the difference between the nested dissection and kd-tree. For me they look same representing a tree data structure for a distribution of points in a multi-dimensional ...
39 views

### Documentation of “bin number” trees

TL;DR: I implemented a special (?) binary tree and can't find any further details on the method I used on the internet. I would like to know if there are any scientific papers discussing my ...
302 views

### Analysis expected depth of a binary search tree given random values?

I have a guess about the problem above. Suppose I have a binary search tree $T$ initially empty. Suppose I drawn $x_1,\ldots,x_k$ (from some real interval $[a,b]$) keys and I want to insert the keys ...
91 views

### Splay trees: why are depths of nodes on the access path halved?

The original paper describing splay trees Self-Adjusting Binary Search Trees by Sleator and Tarjan claims that: Splaying not only moves x to the root, but roughly halves the depth of every node ...
141 views

### Algorithm to find nodes with given a distance

We have a map of a branching river without any islands (tree of confluences, see picture), there are piers on the river bank ( not necessarily on the confluences). We are given the distances between ...
569 views

### Tight upper bound on height of a red-black tree

"Introduction to Algorithms" by Cormen et al., 3rd edition, Lemma 13.1 states that A red-black tree with $n$ internal nodes has height at most $2\log(n+1)$, i.e. $h \le 2\log(n+1)$. Can equality ...
138 views

### Finding post-order traversal of a binary tree from its in-order and pre-order traversals lower bound

I know that we can construct a BST by just having its pre-order traversal in $O(n)$ time (this link). But what if the tree is just a binary tree and we have its in-order and pre-order traversals? I ...
113 views

### What would be the probabilty of a randomly generated tree to be a Red-Black Tree

The question is not related to the homework I was working on a homework, and the specification was to generate a random tree with n elements(n being in the thousands for the assignment) and asked me ...
250 views

### WAVL Trees - Number of rebalances

I was experimenting with a WAVL Tree code I wrote. One thing I noticed is that on average, there was much less rebalancing actions after delete than after insert. In fact, average rebalances after ...
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### How to cut a full binary tree evenly

Given a full binary tree of size $n$, I need to cut the tree into two by choosing an edge and removing it. I want the cutting to be as equal as possible (i.e. the two subtrees have as equal weight as ...
19 views

### How to avoid empty spots when inserting in a B+tree

When you insert a new item in a B+tree leaf node, and the node is full, two new nodes are created and they're both filled halfway and the values get re-distributed accordingly. Before the insertion: ...
634 views

### How to efficiently create balanced KD-Trees from a static set of points

From Wikipedia, KD-Trees: Alternative algorithms for building a balanced k-d tree presort the data prior to building the tree. They then maintain the order of the presort during tree construction ...
612 views

### Red black tree - Insert/Delete proof of correctness

From the Cormen book I was studying the chapter focused on the red black tree. I was particularly interested in why the procedures for insert/delete fixup works (namely a formal proof). I report both ...
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### Can the right childs of a node in an AVL tree be both balanced after an insert and need rebalancing?

I was trying to come up with a case where one would need rebalance the following case in an AVL tree: I think that case impossible to happen during an insert. It seems to me that its impossible to ...
54 views

### Two flavors of red-black trees with different performance

I've implemented two red-black trees (using the pseudo-code in CLRS) with slightly different flavors: 1) In the first tree, all the data is stored in the leaves. 2) In the second tree, the data is ...