Assume non-zero points \(z\) and \(w\) on the Gauss plane. Multiplication by \(z = r e^{i\theta}\) scales the plane by \(r\) and rotates it by \(\theta\).
The Gauss plane under \(w \mapsto z\,w\)
Write \(z = r e^{i\theta}\) and \(w = s e^{i\varphi}\). Since \(e^{i\theta} e^{i\varphi} = e^{i(\theta+\varphi)}\),
$$ z\,w \;=\; (r s)\, e^{\,i(\theta + \varphi)} . $$Thus \(|zw| = r|w|\) and \(\operatorname{Arg}(zw) \equiv \operatorname{Arg} w + \theta \pmod{2\pi}\). Hence multiplication by \(z\) is a rotation about the origin followed by a scaling centred at the origin. The shape of the tree is preserved: every angle stays the same, and every length is multiplied by the same factor \(r\).
In coordinates, with \(z = a + bi\) and \(w = x + yi\),
$$ z\,w = (ax - by) + (ay + bx)\,i, \qquad \begin{pmatrix} ax - by \\[1pt] bx + ay \end{pmatrix} = \begin{pmatrix} a & -b \\[1pt] b & a \end{pmatrix}\begin{pmatrix} x \\[1pt] y \end{pmatrix} = r\begin{pmatrix} \cos\theta & -\sin\theta \\[1pt] \sin\theta & \cos\theta \end{pmatrix}\begin{pmatrix} x \\[1pt] y \end{pmatrix}. $$