#[MATL](https://github.com/lmendo/MATL), <s>13</s> 12 bytes

    ttf2/k)>~4L)

[Try it online!](https://tio.run/##y00syfn/v6QkzUg/W9OuzsRH8///aEMzHQVDEyA20FGw0FEw11Gw1FEw1lEw0lEAicYCAA) Or [verify all test cases](https://tio.run/##dY@xDsMgDER3voI1ElKDgQJLpowZs0WR0qVL2425v07PdqaqlUBYd88@/Lq1Z@@Hae1Ol8cwveMymHntfaujsz7h4s14yFnazVaDyiWrjDJB9lfUUekiTnUWKLpYZWIUTU3QSRxF1OdA7uJRWS3OFyKDSNL0fWDEf8aPHbxkycwgPzgHlJNEOLHBu5Kkh/0D).

An array is truthy if it is non-empty and all its entries are nonzero. Otherwise it's falsy. Here are [some examples](http://matl.tryitonline.net/#code=YCAgICAgICAgICAgICUgaW5maW5pdGUgbG9vcCB0byB0YWtlIGFsbCBpbnB1dHMKPyd0cnV0aHknICAgICUgaWY6IHRydXRoeQp9J2ZhbHN5JyAgICAgJSBlbHNlOiBmYWxzeQpdICAgICAgICAgICAgJSBlbmQgaWYKRFQgICAgICAgICAgICUgZGlzcGxheSBhbmQgZW5kIGxvb3A&input=W10KWzAgMSAyXQpbMyA0IC01XQpbMSAxOyAxIDFd).

###Explanation

    t     % Take input implicitly. Duplicate
    tf    % Duplicate and push indices of nonzero entries. This gives [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