In data structures, a path refers to a sequence of vertices or nodes that are connected by edges. It is often used to describe the route or connection between two specific nodes within a graph or a tree. There are two types of paths commonly used in data structures: simple path and closed path.

## Simple Path

A simple path is a path in which no vertex is repeated. This means that each node in the path appears only once.

In other words, there are no cycles or loops in a simple path. It is also known as a simple cycle-free path.

To better understand this concept, let’s consider an example:

**Graph:**A -> B -> C -> D**Simple Path:**A -> B -> C -> D

In the above example, we can see that the vertices A, B, C, and D form a simple path as there are no repeated vertices.

## Closed Path

A closed path is a type of path that starts and ends at the same vertex. It forms a loop by connecting different nodes but eventually returns to its starting point.

Let’s consider another example to illustrate this concept:

**Graph:**P -> Q -> R -> S -> P**Closed Path:**P -> Q -> R -> S -> P

In this example, the vertices P, Q, R, and S form a closed path as it starts at vertex P and ends at vertex P while covering all other vertices in between.

### Main Differences Between Simple Path and Closed Path:

__Node Repeats:__In a simple path, no node is repeated, whereas in a closed path, the starting and ending node is the same.__Cycles:__Simple paths do not contain cycles or loops, whereas closed paths form loops by connecting different nodes.

Understanding the difference between simple path and closed path is important when working with graphs and trees in data structures. It allows us to analyze and traverse various paths within these structures efficiently.

To summarize, a simple path is a path without any repeated vertices or cycles, while a closed path starts and ends at the same vertex forming a loop. Both types of paths have their significance and applications in data structures.