Balancing Trees

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Questions and Answers

Within the context of binary tree balancing, which of the following statements accurately differentiates between a perfectly balanced tree and a not perfectly balanced tree?

A perfectly balanced tree requires all nodes to have exactly two children, whereas a not perfectly balanced tree allows nodes to have zero or one child.
A perfectly balanced tree necessitates that the subtree heights must differ by at most two for any given node, as opposed to a not perfectly balanced tree where no such constraint applies.
A perfectly balanced tree dictates that the subtrees of each node must be of the same height, while a not perfectly balanced tree allows subtree heights to differ by at most one for at least one node. (correct)
A perfectly balanced tree mandates that all leaf nodes must reside at the identical level, whereas a not perfectly balanced tree permits leaf nodes to exist across multiple levels.

In the context of binary tree rotations, right rotation can be performed on any node irrespective of whether it has a left child.

False (B)

Elaborate on the sequence of node re-positioning involved in executing a right rotation on a designated node within a binary tree.

Move the left child into the node's position. 2. Move the right child of the former left child to become the node's new left child. 3. Move the node to become the right-child of its former left child.

During a left rotation on a node, the node is repositioned to become the ______-child of its former right child.

left Signup and view all the answers

Match the following tree balancing methods with their respective operational characteristics:

Right Rotation = Pivots a node to the right, repositioning its left child upward. Left Rotation = Pivots a node to the left, repositioning its right child upward. Perfectly Balanced Tree = Subtrees of each node exhibit identical height. Not Perfectly Balanced Tree = Subtree heights for at least one node differ by at most one. Signup and view all the answers

What distinguishing characteristic defines an AVL tree in contrast to a generic binary search tree?

AVL trees inherently maintain balance by ensuring that the height difference between the subtrees of any node does not exceed one. (A) Signup and view all the answers

An AVL tree's structural integrity remains uncompromised even if a single node violates the height balance property.

False (B) Signup and view all the answers

Articulate the practical implications of the assertion that 'AVL trees permit some apparently unbalanced trees'.

Certain AVL tree configurations may visually convey an impression of imbalance, yet they adhere to the stipulated height constraint, wherein subtree heights differ by a maximum of one. Signup and view all the answers

Nodes in an AVL tree are categorized as left-heavy, balanced, or right-heavy based on their ______ factor, which indicates the height difference between subtrees.

balancing Signup and view all the answers

Match each AVL tree term with its corresponding definition:

AVL Tree = A height-balanced binary search tree where the height difference between left and right subtrees of any node is at most 1. Balance Factor Violation = Occurs when the height difference between left and right subtrees of a node exceeds 1, necessitating tree rebalancing. Left-Heavy Node = A node in an AVL tree where the height of the left subtree is greater than the height of the right subtree. Right-Heavy Node = A node in an AVL tree where the height of the right subtree is greater than the height of the left subtree. Signup and view all the answers

When a new node insertion into a red-black tree violates its defining properties, which corrective measure is typically employed to restore structural integrity?

Node recoloring and tree rotations. (A) Signup and view all the answers

In a red-black tree, the root node is permitted to be red under specific circumstances.

False (B) Signup and view all the answers

Explain the implications of the red-black tree property mandating that every path from a given node to any of its descendant NULL nodes contains the same number of black nodes.

This property ensures approximate balance in the tree, preventing any path from being more than twice as long as any other, thereby guaranteeing logarithmic time complexity for search, insertion, and deletion operations. Signup and view all the answers

In a red-black tree, a node is designated as 'sentinel' to serve as a convenient means of flagging that you have reached a ______ node.

leaf Signup and view all the answers

Match the following red-black tree properties with their corresponding descriptions:

Every node is either red or black = Establishes the basic color attribute of nodes which is fundamental to balancing. Every leaf (NULL) is black = Ensures that all paths terminate with a black node, contributing to balanced path lengths. If a node is red, then both its children are black = Prevents paths with consecutive red nodes, limiting path length variance. Every simple path from a node to a descendant leaf contains the same number of black nodes = Guarantees approximate balance, ensuring no path is significantly longer than others. Signup and view all the answers

How does a splay tree fundamentally differ from AVL and Red-Black trees in its approach to maintaining balance?

Splay trees leverage amortization to achieve balance over a sequence of operations, while AVL and Red-Black trees adhere to strict height constraints per node. (D) Signup and view all the answers

Splay trees store height data at each node to facilitate efficient rebalancing after each access.

False (B) Signup and view all the answers

Explain the role of 'splaying' operations in maintaining balance and optimizing search times within a splay tree.

Splaying operations move a newly accessed node to the root of the tree through a series of rotations, improving access times for frequently accessed nodes by bringing them closer to the root, and promoting overall balance through redistribution of node depths. Signup and view all the answers

In splay trees, if the accessed node is the 'inside' node of its parent, a ______ rotation (zig-zag or zag-zig) is performed.

double Signup and view all the answers

Match the following splay tree concepts with their corresponding descriptions:

Splaying = The process of moving an accessed node to the root via zig, zig-zag, and zig-zig rotations. Zig Rotation = A single right rotation performed when the accessed node is a left child of the root. Zag Rotation = A single left rotation performed when the accessed node is a right child of the root. Amortized Analysis = The performance characteristic of splay trees where good performance is guaranteed over a sequence of operations. Signup and view all the answers

What is the primary advantage of a splay tree over an AVL tree in scenarios involving non-uniformly distributed access patterns?

Splay trees dynamically adjust node positions to favor frequently accessed nodes closer to the root, leveraging access locality, whereas AVL trees maintain stricter balance regardless of access frequency. (C) Signup and view all the answers

Due to inherent balancing mechanisms, splay trees always result in a tree that is well-balanced after a series of insertions.

False (B) Signup and view all the answers

Contrast the rotation strategies employed in splay trees with those used in AVL trees, emphasizing the implications for overall tree structure and access efficiency.

Splay trees utilize a series of single and double rotations to move accessed nodes to the root, potentially resulting in temporarily unbalanced structures but optimizing access frequency. AVL trees, on the other hand, employ rotations to strictly enforce height balance properties at each node, guaranteeing logarithmic access times at the cost of more frequent rotations. Signup and view all the answers

In the context of rotations within splay trees, a zig-zag rotation is equivalent to a ______ double rotation in an AVL tree.

left-right Signup and view all the answers

Match the following tree types with their respective advantages:

AVL Tree = Guaranteed logarithmic time complexity for all basic operations due to strict height balancing. Red-Black Tree = Good performance with relatively simple implementation; maintains balance through coloring and rotations. Splay Tree = Adaptive to access patterns; frequently accessed nodes are quickly reachable at the cost of potential temporary imbalance. Signup and view all the answers

When is it most appropriate to use a dictionary data structure?

When the data needs to be accessed using a key (C) Signup and view all the answers

In complex data structures, dictionaries cannot be nested within other data structures.

False (B) Signup and view all the answers

Describe an instance where one might utilize a dictionary in conjunction with a tree data structure.

A dictionary could be employed to associate auxiliary data with nodes in a tree, enabling efficient metadata lookup for each node. Signup and view all the answers

The time complexity of looking up a value in a dictionary is typically ______ on average, assuming a good hash function is used.

O(1) Signup and view all the answers

Match the operations commonly associated with dictionaries with the corresponding time complexities:

Insertion = O(1) average, O(n) worst case Deletion = O(1) average, O(n) worst case Lookup = O(1) average, O(n) worst case Signup and view all the answers

Flashcards

Define a Perfectly Balanced Tree

A tree where the sub-trees of each node are of the same height.

Define a Not Perfectly Balanced Tree

A tree where, for at least one node, the heights of the sub-trees differ by at most 1.

What is the Right Rotation Algorithm?

Move left child into Node's position. Move right child of Node's former left child to become the Node's new left child. Move Node to become the right-child of the Node's former left child.

What is the Left Rotation Algorithm?

Move right child into Node's position. Move left child of Node's former right child to become the Node's new right child. Move Node to become the left-child of the Node's former right child.