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

    ttn:2/k)>~4L)

[Try it online!](http://matl.tryitonline.net/#code=dHRuOjIvayk-fjRMKQ&input=WzE2LCAxNCwgMTAsIDgsIDcsIDksIDMsIDIsIDQsIDFd) Or [verify all test cases at once](http://matl.tryitonline.net/#code=YAp0dG46Mi9rKT5-NEwpCkRU&input=WzkwLCAxNSwgMTAsIDcsIDEyLCAyXQpbOTMsIDE1LCA4NywgNywgMTUsIDVdClsxNiwgMTQsIDEwLCA4LCA3LCA5LCAzLCAyLCA0LCAxXQpbMTAsIDksIDgsIDcsIDYsIDUsIDQsIDMsIDIsIDFdClsxMDAsIDE5LCAzNiwgMTcsIDMsIDI1LCAxLCAyLCA3XQpbNSwgNSwgNSwgNSwgNSwgNSwgNSwgNV0KWzQsIDUsIDUsIDUsIDUsIDUsIDUsIDVdCls5MCwgMTUsIDEwLCA3LCAxMiwgMTFdClsxLCAyLCAzLCA0LCA1XQpbNCwgOCwgMTUsIDE2LCAyMywgNDJdClsyLCAxLCAzXQ).

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
    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