Almost! See pesenti's link about definable numbers: most irrationals have no finite representation, although almost all the ones we are familiar with (e.g. sqrt(2), π, e) do: they can be represented as mathematical formulæ. (And indeed, note that such "definable" reals are countable!)

So in my example above, π would not work if, say, your tree was a parse tree of mathematical formulæ: one could locate the formula for π in finite time. Unfortunately I cannot give an example of a single number which does not have such a finite parse tree, as, by its very nature, I cannot write down a finite description of it!