The Pythagoras tree is a plane fractal constructed from squares.The construction of the Pythagoras tree begins with a square. Upon this square are constructed two squares, each scaled down by a linear factor of [math]\frac12\sqrt{2}[/math], such that the corners of the squares coincide pairwise. The same procedure is then applied recursively to the two smaller squares, ad infinitum. This applet shows just 9 steps of the construction. Move the red point to see different shapes.

Can you calculate the area of the Pythagoras tree?