<p>Given an array of strings <code>words</code> (<strong>without duplicates</strong>), return <em>all the <strong>concatenated words</strong> in the given list of</em> <code>words</code>.</p>

<p>A <strong>concatenated word</strong> is defined as a string that is comprised entirely of at least two shorter words (not necessarily distinct)&nbsp;in the given array.</p>

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

<pre>
<strong>Input:</strong> words = [&quot;cat&quot;,&quot;cats&quot;,&quot;catsdogcats&quot;,&quot;dog&quot;,&quot;dogcatsdog&quot;,&quot;hippopotamuses&quot;,&quot;rat&quot;,&quot;ratcatdogcat&quot;]
<strong>Output:</strong> [&quot;catsdogcats&quot;,&quot;dogcatsdog&quot;,&quot;ratcatdogcat&quot;]
<strong>Explanation:</strong> &quot;catsdogcats&quot; can be concatenated by &quot;cats&quot;, &quot;dog&quot; and &quot;cats&quot;; 
&quot;dogcatsdog&quot; can be concatenated by &quot;dog&quot;, &quot;cats&quot; and &quot;dog&quot;; 
&quot;ratcatdogcat&quot; can be concatenated by &quot;rat&quot;, &quot;cat&quot;, &quot;dog&quot; and &quot;cat&quot;.</pre>

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

<pre>
<strong>Input:</strong> words = [&quot;cat&quot;,&quot;dog&quot;,&quot;catdog&quot;]
<strong>Output:</strong> [&quot;catdog&quot;]
</pre>

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

<ul>
	<li><code>1 &lt;= words.length &lt;= 10<sup>4</sup></code></li>
	<li><code>1 &lt;= words[i].length &lt;= 30</code></li>
	<li><code>words[i]</code> consists of only lowercase English letters.</li>
	<li>All the strings of <code>words</code> are <strong>unique</strong>.</li>
	<li><code>1 &lt;= sum(words[i].length) &lt;= 10<sup>5</sup></code></li>
</ul>

class Trie:
    def __init__(self):
        self.branches = defaultdict(Trie)
        self.is_word = False
    
    def insert(self, s):
        cur = self
        for c in s:
            cur = cur.branches[c]
        cur.is_word = True

class Solution:
    def findAllConcatenatedWordsInADict(self, words: List[str]) -> List[str]:
        # sort words by length
        words.sort(key=lambda x: len(x))
        trie = Trie()
        res = []
        
        def check(s, i):
            """
            check if s[i:] can be concatenated by strings in word set
            """
            # base case 
            if i == len(s): 
                return True
            
            # recursion
            cur = trie
            for j in range(i, len(s)):
                # no word in trie has prefix s[i,...,j] 
                if s[j] not in cur.branches:
                    return False
                cur = cur.branches[s[j]]
                # s[i,...,j] in trie and s[j+1:] can be concatenated
                if cur.is_word and check(s, j + 1):
                    return True
            
            return False
        
        
        for s in words:
            if check(s, 0):
                res.append(s)
            else:
                trie.insert(s)
        
        return res
            
        
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