In this article, we will show every tree is planar.

**Tree:** A tree is a simple connected graph that consists of **N – 1** edges where **N** is the number of vertices. There is exactly one path between any two vertices in a tree. A tree consists of no cycle.

## Proof every tree is planar

We can proof it in 2 ways.

### Proof By Euler’s Formula

According to Euler’s Formula,

r = e - v + 2 r: number of regions e: number of edges v: number of vertices

The number of edges in a tree, e = v – 1

r = (v - 1) - v + 2 r = v - 1 - v + 2 r = 1

From the above, we can say that the number of regions in every tree is 1. A graph with one region is always planar since it shows there are no loops. Thus, no possibility for the intersection of edges.

Therefore, we can say that **every tree is planar**.

### Proof By Induction

We need to prove that a tree is planar for all **n >= 1** where **n** is the number of vertices.

**Step 1**

First, we will prove that the statement is true for **n = 1**.**n = 1** means a tree with only 1 vertex and no edges. Thus, **n = 1** is the planar.

**Step 2**

In this step, we will assume that a tree **n = k** is planar.

**Step 3**

In this step, we will prove that if **n = k** is true, then **n = k + 1** is also true

We know **n = k** is planar and consists of **k – 1** edge. The tree **n = k + 1** consists of **k** edges. We can convert tree **n = k to n = k + 1** by adding a new vertex and connecting it to one vertex in tree **n = k**. Adding a new vertex and connecting it to one vertex in the tree cannot result in a loop. For example,

As we can see in the figure, the new vertex will just add a new branch to the tree. Adding a new branch will not affect the planarity of the tree.

Therefore, **n = k + 1** is planar.

We prove that **n = k + 1** is planar if **n = k** is planar. Thus, we can say that the statement every tree is planar is true for all **n >= 1**.

**References**

Euler’s Graph