Awesome problem. There is much to consider, here. Why don't you share with us some of what you have considered?
Thought Question: with a .5 + .25 * 2 + .25 * (1/2) = .5 + .5 + .125 = 1.125 daily expected portfolio growth, are we SURE to get anywhere?
Well, okay, but why did we do that? It's a consistent, well-defined, recursive process.
\(\displaystyle 1.125^{n} = 1,000,000\implies n = 117.2962683\)
Did we need to draw the tree?
Wow, stumped. That's amazing... I need to study more.