#MATL, 13 bytes
ttn:2/k)>~4L)
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An array is truthy if it is non-empty and all its entries are nonzero, or is falsy otherwise. Here are some examples.
###Explanation
t % Take input implicitly. Duplicate
tn: % Duplicate and generate range [1 2 ...n] where n is input size
2/k % Divide by 2 and round down
) % Index into input. Gives array of parents, except for the first entry
>~ % True for entries of the input that don't exceed those in the array of parents
4L) % Discard first entry