You definitely would have more odds than evens in a positive, non-trivial circuit.
k = length of circuit
x = number of odds in circuit
k-x = number of evens in circuit
If the bottom member of the circuit is greater than 1, then
\cfrac{3^x - 2^x}{2^k - 3^x} > 1
3^x - 2^x > 2^k - 3^x
2 \cdot 3^x > 2^k - 2^x
2 \cdot 3^x > 2^x(2^{k-x}-1)
2 \cdot \frac{3}{2}^x > 2^{k-x}-1
x > 0.631 k - 1.27
x > 0.6 k (… for all k \geq 41)
Interested to look at the rest