## tries

<br>

* tries, also called prefix tree, are a variant of n-ary tree in which characters are stored in each node.

* each trie node represents a string (a prefix) and each path down the tree represents a word. note that not all the strings represented by trie nodes are meaningful.

* the root is associated with the empty string.

* the * nodes (null nodes) are often used to indicate complete words (usually represented by a special type of child) or a boolean flag that terminates the parent node.

* a node can have anywhere from 1 through alphabet_size + 1 child.

* can be used to store the entire english language for quick prefix lookup (O(k), where k is the length of the string). they are also widely used on autocompletes, spell checkers, and ip routing (longest prefix matching).

* tries structures can be represented by arrays and maps or trees.

<br>

```python
class Trie:

    def __init__(self):
        self.root = {}

    def insert(self, word: str) -> None:
        node = self.root
        for c in word:
            if c not in node:
                node[c] = {}
            node = node[c]
        node['$'] = None
        
    def match(self, seq, prefix=False):
        node = self.root
        for c in seq:
            if c not in node:
                return False
            node = node[c]
        return prefix or ('$' in node)
        
    def search(self, word: str) -> bool:
        return self.match(word)
        
    def starts_with(self, prefix: str) -> bool:
        return self.match(prefix, True)
```

<br>

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### insertion

<br>

* similar to a bst, when we insert a value to a trie, we need to decide which path to go depending on the target value we insert.
* the root node needs to be initialized before you insert strings.

<br>


---

### search

<br>

* all the descendants of a node have a common prefix of the string associated with that node, so it should be easy to search if there are any words in the trie that starts with the given prefix.
* we go down the tree depending on the given prefix, once we cannot find the child node, the search fails.
* we can also search for a specific word rather than a prefix, treating this word as a prefix and searching in the same way as above.
* if the search succeeds, we need to check if the target word is only a prefix of words in the trie or if it's exactly a word (for example, by adding a boolean flag).

<br>


----

### comparison with hash tables

<br>

* hash table wins in terms of time complexity, as its insert is usually `O(1)` (worst case `O(log(N))` and trie's are `O(M)` (where `M` is the maximum length of a key).
* trie wins in terms of space complexity. both `O(M *N)` in theory, but tries can be much smaller as there will be a lot of words that have similar prefix.