Write a recursive java method that receives a reference to the root of a binary tree that contains characters $($upper and lowercase letters from the English alphabet, digits, and special characters$)$ and then prints and returns the count of all valid passwords constructed from the values in the nodes of the tree when traversed in a pretorder traversal.

Important notes:

$1-$ A password is valid if it is of at least $6$ characters of length and contains at least one lowercase letter, and one uppercase letter, two digits and one of the following special characters $($@$,*,$ $$,!,$ and &$).$

$2-$ Characters of the password must be in the order they appear when the tree is traversed in a pretorder traversal.

$3-$ Your method must find the result with minimum time.

$4-$ You cannot use any other data structure $($array$,$ arraylist, linked list, set, stack, queue, $...$ etc. are all not allowed$).$