## Introduction

The AVL tree is a self-balancing binary search tree. It maintains the height balance property, which ensures that the height difference between the left and right subtrees of any node is at most 1. This property guarantees that the tree remains balanced, resulting in efficient search, insertion, and deletion operations.

## Implementation

To visualize an AVL tree in C#, we need to implement the AVL tree data structure and provide a way to display the tree graphically. Let’s start by implementing the AVL tree class:

```
public class AVLTree
{
private class Node
{
public int Key;
public int Height;
public Node Left;
public Node Right;
public Node(int key)
{
Key = key;
Height = 1;
}
}
private Node root;
// Other methods and properties...
}
```

The AVL tree class consists of a private nested class called Node, which represents a node in the tree. Each node contains a key, height, and references to its left and right child nodes. The AVL tree class also has a private root node variable.

## Insertion

Now, let’s implement the insertion method for the AVL tree. When inserting a new key, we need to maintain the height balance property by performing rotations if necessary. Here’s an example of how to insert a key into the AVL tree:

```
public void Insert(int key)
{
root = Insert(root, key);
}
private Node Insert(Node node, int key)
{
if (node == null)
return new Node(key);
if (key < node.Key)
node.Left = Insert(node.Left, key);
else if (key > node.Key)
node.Right = Insert(node.Right, key);
else
return node; // Key already exists
node.Height = 1 + Math.Max(GetHeight(node.Left), GetHeight(node.Right));
int balance = GetBalance(node);
if (balance > 1 && key < node.Left.Key)
return RotateRight(node);
if (balance < -1 && key > node.Right.Key)
return RotateLeft(node);
if (balance > 1 && key > node.Left.Key)
{
node.Left = RotateLeft(node.Left);
return RotateRight(node);
}
if (balance < -1 && key < node.Right.Key)
{
node.Right = RotateRight(node.Right);
return RotateLeft(node);
}
return node;
}
private int GetHeight(Node node)
{
if (node == null)
return 0;
return node.Height;
}
private int GetBalance(Node node)
{
if (node == null)
return 0;
return GetHeight(node.Left) - GetHeight(node.Right);
}
private Node RotateRight(Node node)
{
Node newRoot = node.Left;
Node temp = newRoot.Right;
newRoot.Right = node;
node.Left = temp;
node.Height = 1 + Math.Max(GetHeight(node.Left), GetHeight(node.Right));
newRoot.Height = 1 + Math.Max(GetHeight(newRoot.Left), GetHeight(newRoot.Right));
return newRoot;
}
private Node RotateLeft(Node node)
{
Node newRoot = node.Right;
Node temp = newRoot.Left;
newRoot.Left = node;
node.Right = temp;
node.Height = 1 + Math.Max(GetHeight(node.Left), GetHeight(node.Right));
newRoot.Height = 1 + Math.Max(GetHeight(newRoot.Left), GetHeight(newRoot.Right));
return newRoot;
}
```

The Insert method takes a key as input and calls the private Insert method to recursively insert the key into the AVL tree. The private Insert method performs the actual insertion and updates the heights of the nodes. It also checks the balance factor of the node and performs rotations if necessary to maintain the height balance property.

## Visualization

To visualize the AVL tree, we can use a graphical library like Windows Forms or WPF to draw the tree on a canvas. Here's an example of how to visualize an AVL tree using Windows Forms:

```
using System;
using System.Drawing;
using System.Windows.Forms;
public class AVLTreeVisualizer : Form
{
private AVLTree avlTree;
public AVLTreeVisualizer(AVLTree tree)
{
avlTree = tree;
Size = new Size(800, 600);
}
protected override void OnPaint(PaintEventArgs e)
{
base.OnPaint(e);
Graphics g = e.Graphics;
DrawNode(g, avlTree.Root, 400, 50, 200);
}
private void DrawNode(Graphics g, AVLTree.Node node, int x, int y, int spacing)
{
if (node == null)
return;
int radius = 20;
int height = 40;
g.FillEllipse(Brushes.White, x - radius, y - radius, 2 * radius, 2 * radius);
g.DrawEllipse(Pens.Black, x - radius, y - radius, 2 * radius, 2 * radius);
g.DrawString(node.Key.ToString(), Font, Brushes.Black, x - radius + 5, y - radius + 5);
if (node.Left != null)
{
int leftX = x - spacing;
int leftY = y + height;
g.DrawLine(Pens.Black, x, y + radius, leftX, leftY - radius);
DrawNode(g, node.Left, leftX, leftY, spacing / 2);
}
if (node.Right != null)
{
int rightX = x + spacing;
int rightY = y + height;
g.DrawLine(Pens.Black, x, y + radius, rightX, rightY - radius);
DrawNode(g, node.Right, rightX, rightY, spacing / 2);
}
}
}
// Usage example:
AVLTree tree = new AVLTree();
tree.Insert(10);
tree.Insert(5);
tree.Insert(15);
tree.Insert(3);
tree.Insert(7);
tree.Insert(12);
tree.Insert(17);
Application.Run(new AVLTreeVisualizer(tree));
```

The AVLTreeVisualizer class extends the Windows Forms Form class and overrides the OnPaint method to draw the AVL tree. The DrawNode method recursively draws each node of the tree using the Graphics object provided by the OnPaint method. The example usage at the end creates an AVL tree, inserts some keys, and displays the tree using the AVLTreeVisualizer class.

## Conclusion

In this article, we have explored how to visualize an AVL tree in C#. We implemented the AVL tree data structure and provided an example of how to insert keys into the tree while maintaining the height balance property. We also demonstrated how to use a graphical library like Windows Forms to visualize the AVL tree graphically. Visualizing data structures can be helpful for understanding their behavior and debugging purposes.