Setting up a conditional string for an array based UL

So I have this code as part of a larger thing I'm working on but this part is the most important, basically I am trying to create an if statement inside the string in PHP where the statement would check the checkbox "checked" value, and if it was positive to show the rest of the children, in a way it will be a kind of lazy load for the child elements since there will be a very large amount of them being parsed, and it would be way faster to not parse them all at once. I tried to get an ajax function working but I am also very new to trying that, full code can be seen here: https://pastebin.com/JX61h1TZ lines 47-61

Any help would be greatly appreciated!

$str .= '
' . '' . '' . $node . '';  [[IF STATEMENT]] $str.=displayHtmlCategoryTree($children, null, $pathSeparator, $currentPath);          }       }$str .= '
'; 

Here is the jQuery I started to loop into it