Markov 2-tuples

In probability theory and intertemporal portfolio choice, the Kelly criterion (or Kelly strategy or Kelly bet), also known as the scientific gambling method, is a formulafor bet sizing that leads almost surely to higher wealth compared to any other strategy in the long run (i.e. approaching the limit as the number of bets goes to infinity). The Kelly bet size is found by maximizing the expected value of the logarithm of wealth, which is equivalent to maximizing the expected geometric growth rate. The Kelly Criterion is to bet a predetermined fraction of assets, and it can seem counterintuitive.

There is no tree. Kelly betting assumes constant returns to sacle. 

So there is one solution regardless of N, the total number of coin tosses. This is the same as guaranteeing the Nyquist sampling rate.  When you compute the betting ratio you are splitting you stash of coin tosses with the random machine.  The Kelly criteria tells you what that split is.

But for Markov 3 and up, you have to be careful as the number of coin tosses,N,  matters. The returns to scale are not constant if you jump the Markov tree. In economics these are the mergerss and  sell offs.