Trees

Beginner’s Guide to Trees in Data Structures: Concepts, Diagrams & Terminology

Diving into DSA: Understanding Trees - Your First Look!

Hey everyone! Welcome back to our journey through the fascinating world of Data Structures and Algorithms. So far, we might have touched upon linear data structures like arrays, linked lists, stacks, and queues, where data is arranged in a sequence. But what happens when data isn't just in a straight line? What if it has a more hierarchical relationship, like an organizational chart or a family tree?

That's where Trees come in! Trees are powerful, non-linear data structures that are super important in computer science. They help us organize data in ways that allow for efficient searching, sorting, and relationships. Think of file systems on your computer, the structure of an HTML page, or even how AI algorithms make decisions – trees are often silently working behind the scenes!

This tutorial is your very first step into understanding trees. We'll start with the basics and gently introduce you to the key ideas.

What Exactly is a Tree (in DSA)?

In the world of Data Structures, a Tree is a collection of entities called nodes connected by links called edges. It's a hierarchical structure where:

There's a special node called the root. This is the topmost node in the tree, and it's the only node that doesn't have a parent.
Every node (except the root) has exactly one parent node. This establishes the hierarchical relationship.
A parent node can have one or more child nodes.
There are no cycles (you can't start at a node and follow edges to get back to the same node). This means there's only one unique path from the root to any other node.

Think of it like an upside-down tree: the root is at the top, and the branches (edges) spread downwards to the leaves (nodes with no children).

This diagram shows a simple general tree structure. Node 'A' at the very top is the root node. Edges connect parent nodes to their children, showing the hierarchy spreading downwards. Notice that node 'C' has two children ('F' and 'G'), which is allowed in a general tree.

Getting Specific: Binary Trees

While general trees can have any number of children per node, a particularly common and important type is the Binary Tree.

A Binary Tree is a tree data structure where each node has at most two children. These children are typically referred to as the left child and the right child. It's this constraint of having at most two children that makes binary trees particularly useful for many algorithms.

Here we see a Binary Tree. The key difference from a general tree is that each node has at most two children (a left child and a right child). For instance, node 'A' has two children, node 'B' has two, and node 'C' has only one. Nodes like 'D', 'E', and 'F' have zero children.

The Building Blocks: Nodes

At the heart of any tree is the Node. A Node is the fundamental element in a tree. It typically contains:

Some data (the actual value or information stored in that node).
References (or pointers) to its children nodes. In a binary tree, a node will have references for its left child and its right child (which might be null if the child doesn't exist).

This diagram illustrates the basic internal structure of a Node in a binary tree. Each node holds some piece of Data and contains references (often implemented as pointers) that point to its Left Child and Right Child. If a child doesn't exist, the corresponding pointer is typically null.

Family Relationships: Parent, Children, Left/Right Child

Parent: The node directly above a given node in the hierarchy is its parent.
Children: The nodes directly connected below a given node are its children.
Left Child: In a binary tree, the child node linked via the left reference.
Right Child: In a binary tree, the child node linked via the right reference.

This diagram highlights the direct relationship between a Parent Node and its Children in a binary tree context. The parent node is connected via edges (shown as arrows pointing downwards) to its Left Child and Right Child.

The Ends of the Branches: Leaf Nodes

Nodes that do not have any children are called Leaf Nodes (or sometimes external nodes). They are the nodes at the very bottom of the tree structure.

This tree diagram shows how to identify Leaf Nodes. Leaf nodes are the ones at the bottom of the tree that have no children (no edges pointing downwards from them). In this diagram, nodes 'D', 'F', and 'H' are the leaf nodes, highlighted in orange.

Tracing Back: Ancestors

The Ancestors of a node are all the nodes on the unique path from the root node to that specific node, excluding the node itself.

Example:
Imagine a tree where the root is node 'A'. 'A' has a child 'B', 'B' has a child 'C', and 'C' has a child 'D'.
For node 'D':

Its parent is 'C'
The parent of 'C' is 'B'
The parent of 'B' is 'A'

So, the ancestors of node 'D' are 'C', 'B', and 'A'.

This diagram illustrates the concept of Ancestors. For the target node 'D', its ancestors are all the nodes on the path from the root ('A') down to 'D', excluding 'D' itself. In this case, the ancestors of 'D' are 'A', 'B', and 'C', which are highlighted along with 'D' to show the path.

A Tree Within a Tree: Subtrees

A Subtree is essentially a tree structure formed by a node and all of its descendants.

Example:
Consider the tree: Root A, A → B, A → C, B → D, B → E, C → F.

The subtree rooted at node 'B' includes node 'B' and its descendants 'D' and 'E'.
The subtree rooted at node 'C' includes node 'C' and its descendant 'F'.
The subtree rooted at the root node 'A' is the entire tree.

A Subtree consists of a node and all of its descendants. The first diagram highlights the subtree rooted at node 'B' (nodes B, D, E). The second diagram highlights the subtree rooted at node 'C' (nodes C, F, G). Each subtree itself forms a valid tree structure. The entire tree is also a subtree rooted at 'A'.

Going Down the Levels: Levels

The Level of a node is its distance from the root node. The root node is typically considered to be at Level 0.

Example:
Using the same tree:

Node 'A' (the root) is at Level 0.
Nodes 'B' and 'C' are at Level 1.
Nodes 'D', 'E', and 'F' are at Level 2.

This diagram shows the Level of each node. The root node ('A') is at Level 0. Nodes directly connected to the root ('B' and 'C') are at Level 1. Their children ('D', 'E', and 'F') are at Level 2, and so on. The level is the number of edges on the path from the root to the node.

How Tall is the Tree? Height

The Height of a tree is defined as the maximum level of any node in the tree.

Example:
Using the same tree:

The maximum level is 2.
Therefore, the height of this tree is 2.

If we add node 'G' as a child of 'D', the path A → B → D → G has 3 edges, so height becomes 3.

The Height of the tree is the maximum level number found among all nodes in the tree. In this diagram, the levels range from 0 up to 4 (at node 'H'). Therefore, the height of this tree is 4. It corresponds to the number of edges on the longest path from the root to a leaf node (e.g., A → C → F → G → H is a path with 4 edges).

Conclusion

Phew! We've covered quite a few terms today: tree, binary tree, node, parent, children, left child, right child, leaf node, ancestor, subtree, level, and height. We looked at examples and diagrams to make these core concepts clearer.

Understanding these fundamental concepts is crucial before we move on to exciting topics like different types of binary trees (like Binary Search Trees!), how to traverse trees (visiting each node), and how trees are used to solve real-world problems.

Stay tuned for the next tutorial where we'll explore more about Binary Search Trees and why they are so incredibly useful!