ACM ICPC Amritapuri Regional 2012 - Problem H

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I cannot get any idea to solve this problem? I only proved that the expected value of min(x,y) = 1/3 and max(x,y) = 2/3 when x and y in the range [0,1] using double integrals. However, I cannot get any idea to solve the rest of the problem when many min-max branches are involved. Thanks in advance.

I will give you a hint:

For every node, calculate a polynomial p(x), which gives the probability of the value of node being < x.
You can calculate this if you know the probability polynomial for its children.

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Actually I got the polynomial of min and max:
prob of min(x,y) = ( x^2 + y^2 ) / 2 - ( x^3 + y^3 ) / 3
prob of max(x,y) = ( x^3 + y^3 ) / 3

But after trying to generalize when getting results from children, I get wrong answer for the third test case

Can you please elaborate your solution more?