See Barnsley's Fern by Eric W. Weisstein from MathWorld, a Wolfram Web Resource.

The program starts from a random point. At each step, it randomly picks a function (with non-uniform probability) and applies the function to the point to find the next point. It then plots that point.

Each function has the form:

    X(n+1) = A * X(n) + B * Y(n) + C
    Y(n+1) = D * X(n) + E * Y(n) + F

The Form_Load event handler initializes the parameters for each function in the m_Func array, and the probability for each function. It also sets the color to be used when plotting a point found by each function. Using different colors is interesting and gives you some sense of what each set of functions does but the final result here is drawn in green.

Private m_Prob(3) As Single
Private m_Func(3, 1, 1) As Single
Private m_Plus(3, 1) As Single
Private m_Clr(3) As Color

Private Sub Form1_Load(ByVal sender As System.Object, ByVal _
    e As System.EventArgs) Handles MyBase.Load
    ' Initialize the fern functions.
    m_Clr(0) = Color.Red
    m_Prob(0) = 0.01
    m_Func(0, 0, 0) = 0
    m_Func(0, 0, 1) = 0
    m_Func(0, 1, 0) = 0
    m_Func(0, 1, 1) = 0.16
    m_Plus(0, 0) = 0
    m_Plus(0, 1) = 0

    m_Clr(1) = Color.Green
    m_Prob(1) = 0.85
    m_Func(1, 0, 0) = 0.85
    m_Func(1, 0, 1) = 0.04
    m_Func(1, 1, 0) = -0.04
    m_Func(1, 1, 1) = 0.85
    m_Plus(1, 0) = 0
    m_Plus(1, 1) = 1.6

    m_Clr(2) = Color.Blue
    m_Prob(2) = 0.08
    m_Func(2, 0, 0) = 0.2
    m_Func(2, 0, 1) = -0.26
    m_Func(2, 0, 1) = -0.23
    m_Func(2, 1, 1) = 0.22
    m_Plus(2, 0) = 0
    m_Plus(2, 1) = 1.6

    m_Clr(3) = Color.White
    m_Prob(3) = 0.06
    m_Func(3, 0, 0) = -0.15
    m_Func(3, 0, 1) = 0.28
    m_Func(3, 1, 0) = 0.26
    m_Func(3, 1, 1) = 0.24
    m_Plus(3, 0) = 0
    m_Plus(3, 1) = 0.44

    m_Clr(0) = Color.Lime
    m_Clr(1) = Color.Lime
    m_Clr(2) = Color.Lime
    m_Clr(3) = Color.Lime

    MakeFern()
    picCanvas.Invalidate()
End Sub

Subroutine MakeFern starts at point (1, 1) and repeatedly applies randomly selected functions, plotting each point on the Bitmap.

Private m_Bm As Bitmap

Private Sub MakeFern()
    m_Bm = New Bitmap(picCanvas.ClientSize.Width, _
        picCanvas.ClientSize.Height)
    Dim gr As Graphics = Graphics.FromImage(m_Bm)

    gr.Clear(picCanvas.BackColor)

    Dim rnd As New Random
    Dim func_num As Integer
    Dim clr As Color
    Dim x As Single = 1
    Dim y As Single = 1
    Dim x1 As Single
    Dim y1 As Single
    Dim ix As Integer
    Dim iy As Integer
    For i As Long = 1 To 100000
        Dim num As Double = rnd.NextDouble()
        For j As Integer = 0 To 3
            num = num - m_Prob(j)
            If num <= 0 Then
                func_num = j
                clr = m_Clr(j)
                Exit For
            End If
        Next j

        x1 = x * m_Func(func_num, 0, 0) + y * _
            m_Func(func_num, 0, 1) + m_Plus(func_num, 0)
        y1 = x * m_Func(func_num, 1, 0) + y * _
            m_Func(func_num, 1, 1) + m_Plus(func_num, 1)
        x = x1
        y = y1

        Const W_XMIN As Single = -4
        Const W_XMAX As Single = 4
        Const W_YMIN As Single = -0.1
        Const W_YMAX As Single = 10.1
        Const W_WID As Single = W_XMAX - W_XMIN
        Const W_HGT As Single = W_YMAX - W_YMIN
        ix = (x - W_XMIN) / W_WID * _
            picCanvas.ClientSize.Width
        iy = (picCanvas.ClientSize.Height - 1) - (y - _
            W_YMIN) / W_HGT * picCanvas.ClientSize.Height
        If ix >= 0 AndAlso iy >= 0 _
            AndAlso ix < picCanvas.ClientSize.Width _
            AndAlso iy < picCanvas.ClientSize.Height _
        Then
            m_Bm.SetPixel(ix, iy, clr)
        End If
    Next i
End Sub