<p>You are given a positive integer <code>n</code>.</p>

<p>A binary string <code>x</code> is <strong>valid</strong> if all <span data-keyword="substring-nonempty">substrings</span> of <code>x</code> of length 2 contain <strong>at least</strong> one <code>&quot;1&quot;</code>.</p>

<p>Return all <strong>valid</strong> strings with length <code>n</code><strong>, </strong>in <em>any</em> order.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">n = 3</span></p>

<p><strong>Output:</strong> <span class="example-io">[&quot;010&quot;,&quot;011&quot;,&quot;101&quot;,&quot;110&quot;,&quot;111&quot;]</span></p>

<p><strong>Explanation:</strong></p>

<p>The valid strings of length 3 are: <code>&quot;010&quot;</code>, <code>&quot;011&quot;</code>, <code>&quot;101&quot;</code>, <code>&quot;110&quot;</code>, and <code>&quot;111&quot;</code>.</p>
</div>

<p><strong class="example">Example 2:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">n = 1</span></p>

<p><strong>Output:</strong> <span class="example-io">[&quot;0&quot;,&quot;1&quot;]</span></p>

<p><strong>Explanation:</strong></p>

<p>The valid strings of length 1 are: <code>&quot;0&quot;</code> and <code>&quot;1&quot;</code>.</p>
</div>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= n &lt;= 18</code></li>
</ul>

class Solution:
    def validStrings(self, n: int) -> List[str]:
        def is_valid(s):
            for i in range(len(s) - 1):
                if s[i] == '0' and s[i+1] == '0':
                    return False
            return True

        def generate_strings(n):
            if n == 1:
                return ["0", "1"]
            prev_strings = generate_strings(n-1)
            valid_strings = []
            for s in prev_strings:
                if is_valid(s + "0"):
                    valid_strings.append(s + "0")
                if is_valid(s + "1"):
                    valid_strings.append(s + "1")
            return valid_strings

        return generate_strings(n)
80