1 Balanced Binary Search Trees
... Note that inserts happen only at a new leaf. Also an insertion may create an imbalance only at
the vertices in the path from the new leaf to the root and nowhere else.
The local operation which are used to maintain balance after insertion and deletion is rotation. This operation moves the nodes arou ...
Binary Trees: Notes on binary trees
... levels are full with exception to the last level
and it is filled from left to right.
A full binary tree is one where if a node has a
child, then it has two children.
Augmenting Data Structures 2
... Maintain a dynamic set of elements with each element x
containing an interval int[x]
Support the following operations:
• INSERT(T,x) : Adds an element x whose int field contains
an interval to the tree
• DELETE(T,x): Removes the element x from the tree T
• SEARCH(T,i): Returns a pointer to an el ...
Problem 7—Skewed Trees Trees are particularly annoying to test
... substandard and cannot be guaranteed to avoid the greenery. Hal is not actually afraid of trees—he was evidently born without
fear and could never be called “yellow”—but they are annoying.
Hal's spent enough time observing trees to notice that some are more even than others. Some have branches evenl ...
Binary Search Trees
... – If the tree is a linear chain of n nodes, however,
the same operations take (n) worst-case time.
... necessary by swapping with its parent until it is not
less than its parent
See Figures 10.17 and 10.18
To delete a node from a heap, replace it with the last
node on the bottom level and bubble that node up or
down as necessary
See Figures 10.19 and 10.20
Prelim 1 solutions - Cornell Computer Science
... follow the path up to the root from the inserted or deleted leaf and add or subtract one at each node. This
takes only O(log n) extra time. On a split or join in a 2-3 tree or a rotate in a red-black tree, the new
counts of the affected nodes can be computed locally in O(1) time from the counts of t ...
Lecture 3: Red-black trees. Augmenting data structures
... Lecture 3: Red-black trees. Augmenting data structures
A red-black tree is a binary search tree with the following properties:
1. Every node is either red or black.
2. The root is black.
3. Every leaf (NIL) is black.
4. If a node is red, then both its children are black. Hence there cannot be two co ...
Data Structures in Java
... • Read left child, then parent, then right child
• Essentially scans whole tree from left to right
• inorder(node x)
Binary Trees - Wellesley College
... path from n to a leaf below it. E.g., node G has
height 1, node C has height 2, and node F has
The depth of a node n is the length of the path
from n to the root. E.g., node F has depth 0,
node C has depth 1, and node G has
A binary tree is height-balanced iff at every
node n, t ...
In computer science, an interval tree is a tree data structure to hold intervals. Specifically, it allows one to efficiently find all intervals that overlap with any given interval or point. It is often used for windowing queries, for instance, to find all roads on a computerized map inside a rectangular viewport, or to find all visible elements inside a three-dimensional scene. A similar data structure is the segment tree.The trivial solution is to visit each interval and test whether it intersects the given point or interval, which requires O(n) time, where n is the number of intervals in the collection. Since a query may return all intervals, for example if the query is a large interval intersecting all intervals in the collection, this is asymptotically optimal; however, we can do better by considering output-sensitive algorithms, where the runtime is expressed in terms of m, the number of intervals produced by the query. Interval trees have a query time of O(log n + m) and an initial creation time of O(n log n), while limiting memory consumption to O(n). After creation, interval trees may be dynamic, allowing efficient insertion and deletion of an interval in O(log n). If the endpoints of intervals are within a small integer range (e.g., in the range [1,...,O(n)]), faster data structures exist with preprocessing time O(n) and query time O(1+m) for reporting m intervals containing a given query point.