Data Structures Deep Dive (4/8): Trees: Hierarchical Data Representation

14 min readAug 16, 2023

The Backbone of Hierarchical Data Modeling

Before diving into trees, ensure you’ve grasped the fundamentals of Stacks and Queues: Managing Data Efficiently. It sets the foundation for understanding hierarchical structures like trees!

Introduction to Trees

Trees are one of the most versatile and fundamental data structures. At its core, a tree is a collection of entities, often called nodes, connected by edges that signify a relationship between them. Unlike arrays or linked lists, which are linear structures, trees are hierarchical. This means that each node in a tree, except the root node, is connected to exactly one other node, its parent. This simple yet powerful structure allows trees to represent data in a way that mirrors many real-world scenarios.

Definition of Trees

A tree is defined as a collection of nodes where each node is a data element. The nodes are connected in a way that reflects a hierarchy. There’s a special node, called the root, from which all other nodes stem. Each node can have zero or more child nodes, and a node that has no children is referred to as a leaf node.

Here’s a simple breakdown of the terminology associated with trees:

Root: The topmost node in a tree.
Child: A node directly connected to another node when moving away from the root.
Parent: The converse notion of a child.
Leaf: A node with no children.
Sibling: Nodes with the same parent.

Significance of Trees in Data Structures

Trees play a pivotal role in computer science for several reasons:

Hierarchical Representation: Trees naturally represent hierarchical structures, making them ideal for scenarios where data has parent-child relationships, such as organizational charts or file systems.
Efficient Data Access: Trees, especially balanced ones, allow for faster data access, insertion, and deletion compared to linear data structures. This efficiency is evident in structures like Binary Search Trees (BST).
Flexibility: Trees can expand and shrink easily, making them dynamic and adaptable to various operations.
Diverse Variants: There are multiple types of trees, each tailored for specific applications, such as AVL trees for balancing, Trie trees for word searches, and B-trees for database storage.

Trees Representing Hierarchical Structures

The hierarchical nature of trees makes them the go-to data structure for various applications:

File Systems: Directories and files in a computer’s file system are organized as a tree. The root directory is the tree’s root, and each file or sub-directory is a node.
Organizational Charts: In businesses, an org chart displays the structure of an organization in terms of rank. The CEO sits at the top (root), with various levels of management branching out below.
Document Object Model (DOM): Web pages are structured as trees, with each tag representing a node in the DOM tree.
Game Trees: In artificial intelligence, trees represent possible moves in a game, with each node being a potential state of the game.

In essence, whenever there’s a need to represent data in a hierarchical manner, trees come into play. Their ability to model real-world structures and scenarios is a testament to their significance in data structures and computer science as a whole.

Understanding the Basics of Trees

At its core, a tree in computer science is not much different from the trees we see in nature. Just as a tree grows upwards from its roots, branching out into various limbs and leaves, a data tree starts with a root node and expands into various branches and nodes. Each node in the tree holds data and can have multiple child nodes, but only one parent node, except for the root, which has none.

How Trees Work

When data is added to a tree, it’s placed within a new node. Where this node is added depends on the type of tree and its rules. For instance, in a binary tree, each node can have at most two children. If the data is less than the parent node, it’s placed to the left; if more, to the right.

Removing data involves finding the node and then rearranging the tree so that the hierarchical structure remains intact.

Searching in trees, especially balanced ones, is faster than in a linear data structure like an array or linked list. This is because, with each step, you eliminate half of the remaining nodes from consideration.

Why Trees Matter

Trees are crucial in computer science because they provide a means to represent data hierarchically and allow for efficient data operations like search, insert, and delete. Their non-linear structure means that they can represent complex relationships and structures, from file systems on a computer to the organization of a book’s content.

Components of Trees

Nodes: The fundamental units that hold data in a tree. Every tree consists of one or more nodes.
Root: The topmost node in a tree. It serves as the starting point and has no parent.
Leaves: The nodes in a tree that have no children. They represent the endpoints of a tree’s branches.
Edges: The links that connect nodes in a tree. An edge connects two nodes, signifying a relationship between them.
Parent and Children: In the context of trees, a node that has one or more nodes stemming from it is termed its children, while the node from which a particular node stems is termed its parent. The root is the only exception, as it has no parent.

This diagram represents a simple tree with a root node branching out to two child nodes. One of the child nodes further branches out to two leaf nodes, while the other child node branches to one leaf node.

Depth of a Node:

Definition: The depth of a node refers to the number of edges on the path from the root node to the node in question.
Example: In a tree where the root is labeled A, and it has a child B, which in turn has a child C, the depth of A is 0, the depth of B is 1, and the depth of C is 2.

Height of a Tree:

Definition: The height of a tree is the length of the path from the root to the deepest node in the tree. In other words, it’s the maximum depth of any node in the tree.
Example: Using the same tree as below, where A is the root, and C is the deepest node, the height of the tree is 2 (the depth of node C).

Levels of a Tree:

Definition: The level of a node represents its distance from the root. It’s essentially the depth of the node plus 1, as we start counting levels from 1 (with the root being at level 1).
Example: In our tree, node A is at level 1, node B is at level 2, and node C is at level 3.

Visual representation of a tree with depth and levels

In this diagram:

The root node A has a depth of 0 and is at level 1.
Nodes B and E are children of the root. They have a depth of 1 and are at level 2.
Nodes C and D are children of node B. They have a depth of 2 and are at level 3.

The height of this tree is 2, as the maximum depth of any node (C or D) is 2.

By understanding the depth, height, and levels of a tree, one can gain insights into the structure and characteristics of the tree, which is crucial when performing various operations or algorithms on trees.

While we’ve just scratched the surface of trees, there’s so much more to explore. If you’re finding this journey intriguing, consider following me

Types of Trees

1. Binary Trees

A binary tree is a tree structure in which each node has at most two children, which are referred to as the left child and the right child.

Example: Consider a simple family hierarchy where each person has at most two children.

Binary trees are one of the simplest, yet most powerful data structures. They have only two children, left and right. The maximum number of nodes on level ‘l’ is 2^(l-1). The depth of a tree is the maximum level where a node is present.

Usage: Binary trees are foundational for binary search trees and binary heaps. They are used in many algorithms to achieve efficient searching, sorting, and dynamic data retrieval.

2. Binary Search Trees (BST)

A BST is a binary tree where nodes have a property where all nodes to the left are less than the current node and all nodes to the right are greater.

Example: A tree of numbers where each number is positioned based on its value.

In a BST, for every node, all elements in the left subtree are less, and all elements in the right subtree are greater than the node. This property provides an advantage in searching operations. The in-order traversal of a BST retrieves elements in sorted order.

Usage: BSTs are used in many search applications where data is constantly entering and leaving, like in certain dictionary implementations.

3. AVL Trees

An AVL tree is a self-balancing binary search tree where the difference between the heights of left and right subtrees cannot be more than one for all nodes.

Example: A tree that re-balances itself after every insert or delete operation.

Named after its inventors Adelson-Velsky and Landis, AVL trees maintain their balance (i.e., height-balanced) by ensuring that the difference between the depths of the left and right subtrees is at most 1. Whenever this property is violated, rebalancing is done using tree rotations.

Usage: AVL trees are used in situations where we need to keep the tree balanced, like in certain database systems where balanced search trees are required for performance.

4. Red-Black Trees

A red-black tree is a kind of self-balancing binary search tree where each node has a color (either red or black) and it satisfies certain properties to maintain balance during insertions and deletions.

Example: A tree that colors its nodes to ensure balance.

Red-Black trees ensure that the tree remains approximately balanced, which guarantees O(log n) time for the basic operations (add, delete, find). The tree satisfies the following properties:

Every node is either red or black.
The root is black.
All leaves are black.
If a node is red, then both its children are black.
Every simple path from a node to a descendant leaf has the same black depth (the number of black nodes).

Usage: Red-Black trees are used in many programming languages like C++, Java in their collection libraries to ensure balanced trees.

5. B-Trees

B-trees are balanced search trees designed to work well on direct-access storage devices such as magnetic disks.

Example: A tree that can have more than two children.

B-Trees are generalizations of binary search trees that can have more than two children. They are optimized for systems that read and write large blocks of data. B-Trees guarantee logarithmic insert and delete times and are crucial in database systems where large blocks of data are read/written on disks.

Usage: B-Trees are widely used in databases and filesystems to ensure that data blocks are read/written in a minimum number of disk I/O operations.

6. Trie (Prefix Trees)

A trie is a tree-like data structure used to store a dynamic set or associative array where the keys are usually strings.

Example: A tree that represents a dynamic set of strings.

Tries are an efficient way to represent a set of strings. Each edge of the trie represents a character of a string, and strings share the structure of the trie which allows for efficient storage. Searching, inserting, and deleting operations have a time complexity of O(k), where k is the length of the string.

Usage: Tries are particularly used in applications involving a large dataset of strings, like in search engines, text databases, and for implementing features like auto-suggest.

7. Splay Trees

A Splay Tree is a self-adjusting binary search tree with the additional property that recently accessed elements are quick to be accessed again. Splaying is the process of moving a node to the root after it’s accessed. This is done using a series of tree rotations.

Advantages:

Amortized bounds for operations are good, even if individual operations can be expensive.
Splay Trees do not need to store any bookkeeping data.
Offers a form of cache-obliviousness, as recently accessed elements are kept near the top.

Use Cases: Useful in applications where certain data items are accessed more frequently, and we want to optimize for those frequent accesses.

Example: Consider a BST with nodes 10, 20, 30, 40, 50. If we access the node with the value 30 in a Splay Tree, after the access, the tree will adjust itself to move the 30 node to the root.

Splay trees have a unique property where after every access (whether it’s an insertion, deletion, or a simple search), the node is moved to the root of the tree using a splaying operation. This ensures that frequently accessed nodes are always near the root and can be accessed quickly.

8. Heap

A heap is a specialized tree-based data structure that satisfies the heap property. It can be visualized as a binary tree, but it’s important to note that all binary trees are not heaps.

Types: There are two main types of heaps:

Max-Heap: For any given node I, the value of I is greater than or equal to the values of its children. This property ensures that the largest element is found at the root.
Min-Heap: For any given node I, the value of I is less than or equal to the values of its children. This ensures that the smallest element is at the root.

Heaps are usually implemented as binary trees, but they don’t have to be binary. The important thing about heaps is that they are complete trees. The tree is filled on all levels except possibly the lowest, which is filled from the left up to a point. This makes them suitable to be stored in an array without any gaps. The parent-child relationship can be defined using array indices.

For a given element at position i in the array:

Its Left Child will be at position 2*i + 1
Its Right Child will be at position 2*i + 2
Its Parent Node will be at position (i-1)/2

Usage: Heaps are widely used in algorithms like heap sort and in data structures like priority queues. It’s especially useful in algorithms requiring the highest or lowest element in real time. For instance, the heapq module in Python’s standard library provides functions for creating heaps and manipulating them.

Adding the heap to our list provides a comprehensive overview of tree-like structures that are fundamental in computer science and various applications.

Use Cases and Advantages of Trees

Trees, as a fundamental data structure, have found their way into a multitude of applications. Their hierarchical nature and inherent properties make them suitable for various tasks in computer science and other fields. Here’s a deeper dive into some of the primary use cases and advantages of trees:

1. Hierarchical Data Representation

File Systems: One of the most common uses of trees, especially binary trees, is in the representation of file systems. Each directory can be seen as a node, and each file or sub-directory within it as a child. This structure allows for efficient file searching, organization, and storage management.

Organizational Structures: In businesses and institutions, trees can represent organizational hierarchies. The CEO or president is the root, and each subsequent level of management and employees form the subsequent levels of the tree. This visual representation aids in understanding roles, responsibilities, and chains of command.

2. Efficient Search Operations in BSTs

Binary Search Trees (BSTs): BSTs are a type of binary tree where each node has a value, and all nodes in the left subtree are less than the root, and all nodes in the right subtree are greater. This property ensures that search operations, on average, take logarithmic time, making them highly efficient.

3. Priority Queues Using Heaps

Heaps: Heaps are a special kind of tree used to implement priority queues. In a max heap, for any given node I, the value of I is greater than or equal to the values of its children. This property ensures that the highest (or lowest, in the case of a min-heap) priority element is always at the root, allowing for efficient maximum (or minimum) value retrieval.

4. Dynamic Sorting and Indexing

Trees, especially self-balancing trees like AVL or Red-Black trees, are used in databases for indexing. They allow for dynamic sorting, ensuring that data remains ordered even as new entries are added. This property speeds up search operations and ensures efficient data retrieval.

5. Balancing Load in Network Routing

Network Routing: Trees can be used in network structures to balance load and ensure efficient data packet routing. By understanding the hierarchy and paths within the tree, network routers can make intelligent decisions about where to send data to ensure it reaches its destination quickly and efficiently.

Trees, with their versatile nature, continue to be a cornerstone in the world of data structures. Their adaptability and efficiency in handling various tasks make them indispensable in both classic and modern computing scenarios.

Conclusion

Navigating the intricate branches of data structures, trees stand tall as one of the most versatile and foundational elements in computer science. From simplifying complex hierarchical data to optimizing search operations, trees have proven their mettle time and again. Their adaptability extends beyond mere theory, finding practical applications in everyday technologies, from our file systems to the very infrastructure of the internet. As we’ve journeyed through the depths of trees, their types, and their myriad applications, it’s evident that their roots run deep in the world of algorithms and computing. Whether you’re a budding programmer or a seasoned developer, understanding trees and their nuances can be a game-changer, opening doors to efficient problem-solving and innovative solutions. As we branch out into the future of technology, the knowledge and application of trees will undoubtedly continue to flourish, making our digital forests richer and more robust.

Wrapping up our discussion on trees, I’m eager to introduce the next topic. Join me in the upcoming article as we explore Hash Tables: Quick Data Retrieval. Discover how hash tables revolutionize data storage and offer rapid retrieval. Stay with me for this enlightening continuation of the data structures journey!