Hi Noct,
My comment was not meant to say your answer is wrong.
I was commenting, rather, about the OP:
First, I would say it's not a well posed problem.
Consider the significant differences among these three statements.
[1] Each year it continues to grow at the rate of double the previous years height.
[2] Each year its height changes by a multiplicative factor equal in magnitude to two times the current height of the tree expressed in cm.
[3] Each year its height increases by an amount equal to two times its current height.
Statement [1] is from the OP and is not well posed.
Statement [2] is well posed, and would be, to my thinking, what the OP tried to say.
Statement [3] is well posed and leads to your answer.
If you take the OP as a formula for calculating next year's growth rate from this year's height:
[1] leaves you in the dark as to what units to use: cm, inches, feet, meters and furlongs all give different results.
[2] works, but isn't what the OP says.
[3] tells you how the height changes, but it isn't what the OP says; it doesn't address growth rate at all - rather a growth amount.
Second, a growth rate most commonly refers to the derivative of height with respect to time, and it has the units of [length]/[time].
If [2] is what is meant, calling a factor a rate is incorrect; they have different dimensions. Rate is dimensionless.
If [3] is what is meant, calling a height increase a rate is also incorrect. Summarizing,
Rate = [length/time]; height and height increase = [length] and factor = [dimensionless].
To say Rate = 2 x Height creates a dimensionality error.
To say Factor = 2 x Height also creates a dimensionality error.
To say Height increase = 2 x Height is dimensionally correct, but the OP purports to specify a rate.
So to my mind the OP does not say clearly how the tree grows.
Regarding my comment about alternative, I'm simply saying that if you abandon the idea of calculating a growth rate [or factor or height increase] from the tree's current height, you're pretty much left with taking the OP to say that the height, itself, doubles each year.
