Let's walk through the construction of the **R-way trie** for the set of words: ``` now is the time for all good people to come to the aid of ``` **Assumptions:** - Only lowercase letters and spaces in the input. - Each node can have up to 27 children: 26 for 'a'..'z', 1 for space (if you wish; usually, words are separated, not included in the trie). - Each key is a word, not a sentence; so we insert: ``` now, is, the, time, for, all, good, people, to, come, to, the, aid, of ``` **Order of Insertion:** 1. now 2. is 3. the 4. time 5. for 6. all 7. good 8. people 9. to 10. come 11. to 12. the 13. aid 14. of **Trie Construction (Step by Step):** Let's build the trie **by hand** and draw it in text form. --- We'll use the following format: ``` (root) |-- n -- o -- w |-- i -- s |-- t -- h -- e | |-- i -- m -- e | |-- o |-- f -- o -- r |-- a -- l -- l | |-- i -- d |-- g -- o -- o -- d |-- p -- e -- o -- p -- l -- e |-- c -- o -- m -- e |-- o -- f ``` **Let's explain how this is built:** 1. **Insert "now":** ``` (root) |-- n -- o -- w ``` 2. **Insert "is":** ``` (root) |-- n -- o -- w |-- i -- s ``` 3. **Insert "the":** ``` (root) |-- n -- o -- w |-- i -- s |-- t -- h -- e ``` 4. **Insert "time":** - Shares 't' with "the": ``` (root) |-- n -- o -- w |-- i -- s |-- t -- h -- e | |-- i -- m -- e ``` 5. **Insert "for":** ``` (root) |-- n -- o -- w |-- i -- s |-- t -- h -- e |-- i -- m -- e |-- f -- o -- r ``` 6. **Insert "all":** ``` (root) |-- n -- o -- w |-- i -- s |-- t -- h -- e |-- i -- m -- e |-- f -- o -- r |-- a -- l -- l ``` 7. **Insert "good":** ``` (root) |-- n -- o -- w |-- i -- s |-- t -- h -- e |-- i -- m -- e |-- f -- o -- r |-- a -- l -- l |-- g -- o -- o -- d ``` 8. **Insert "people":** ``` (root) |-- n -- o -- w |-- i -- s |-- t -- h -- e |-- i -- m -- e |-- f -- o -- r |-- a -- l -- l |-- g -- o -- o -- d |-- p -- e -- o -- p -- l -- e ``` 9. **Insert "to":** - Shares 't' with "the" and "time": ``` (root) |-- n -- o -- w |-- i -- s |-- t -- h -- e |-- i -- m -- e |-- o |-- f -- o -- r |-- a -- l -- l |-- g -- o -- o -- d |-- p -- e -- o -- p -- l -- e ``` 10. **Insert "come":** ``` (root) |-- n -- o -- w |-- i -- s |-- t -- h -- e |-- i -- m -- e |-- o |-- f -- o -- r |-- a -- l -- l |-- g -- o -- o -- d |-- p -- e -- o -- p -- l -- e |-- c -- o -- m -- e ``` 11. **Insert "to" again:** - Already present, so nothing changes. 12. **Insert "the" again:** - Already present, so nothing changes. 13. **Insert "aid":** ``` (root) |-- n -- o -- w |-- i -- s |-- t -- h -- e |-- i -- m -- e |-- o |-- f -- o -- r |-- a -- l -- l |-- i -- d |-- g -- o -- o -- d |-- p -- e -- o -- p -- l -- e |-- c -- o -- m -- e ``` 14. **Insert "of":** ``` (root) |-- n -- o -- w |-- i -- s |-- t -- h -- e |-- i -- m -- e |-- o |-- f -- o -- r |-- a -- l -- l |-- i -- d |-- g -- o -- o -- d |-- p -- e -- o -- p -- l -- e |-- c -- o -- m -- e |-- o -- f ``` --- ### Final **R-way trie** (without null links): ``` (root) |-- n -- o -- w |-- i -- s |-- t -- h -- e | |-- i -- m -- e | |-- o |-- f -- o -- r |-- a -- l -- l | |-- i -- d |-- g -- o -- o -- d |-- p -- e -- o -- p -- l -- e |-- c -- o -- m -- e |-- o -- f ``` - Each level represents a character in the key. - Words that share a prefix share nodes up to the divergence. - **No null links are drawn.** - If you wish, you can mark word ends with a `*`, but the problem does not ask for this. --- **If you want to draw this as a tree, you can use indentation or a diagram as above.** Let me know if you need this as an ASCII art tree or a diagram!

**1. Introduction** In computer science, a **trie** (also known as a prefix tree) is a tree data structure used for efficient retrieval of key sequences, typically strings. Each node in a trie represents a common prefix shared among some subset of the keys, and edges correspond to characters. A key characteristic of tries is that they facilitate fast search, insert, and prefix-matching operations, often in time proportional to the length of the key rather than the number of keys stored. An **R-way trie** generalizes the trie concept by allowing each node to have up to R children, where R equals the size of the alphabet. For example, if the alphabet consists of lowercase English letters, R=26; if spaces are included, R=27. This structure is useful in applications such as autocomplete, spell checking, and IP routing. In constructing a trie from a set of keys, we insert each key character by character, creating new nodes as needed. When a key is fully inserted, we mark the terminal node to indicate the end of a valid key. Notably, in the visualization, null links (non-existent children) are omitted for clarity. **Supporting Explanation:** Understanding how to build and visualize a trie is critical for grasping its benefits—particularly, efficient prefix-based lookups. Constructing the trie step-by-step from a list of words demonstrates the sharing of common prefixes, which is the core advantage of trie structures in reducing redundancy and improving search times. --- **2. The Complete Code** Below is a Python implementation that constructs and displays an R-way trie for the given set of words. The code uses a nested dictionary approach for simplicity. ```python # Define the alphabet (lowercase letters plus space) import string ALPHABET = list(string.ascii_lowercase) + [' '] CHAR_TO_INDEX = {ch: idx for idx, ch in enumerate(ALPHABET)} # Function to insert words into the trie def insert_word(trie, word): node = trie for ch in word: if ch not in node: node[ch] = {} node = node[ch] # Mark the end of a word node['$'] = True # Using '$' to denote end of word # Function to display the trie structure def print_trie(node, prefix=''): for ch in sorted(node.keys()): if ch == '$': continue print(f"{prefix}|-- {ch}") print_trie(node[ch], prefix + " ") # List of words to insert words = [ "now", "is", "the", "time", "for", "all", "good", "people", "to", "come", "aid", "of" ] # Build the trie trie = {} for word in words: insert_word(trie, word) # Display the trie print("R-way trie structure:") print_trie(trie) ``` **Explanation of the Code:** - The code defines an alphabet consisting of lowercase letters and space. - It constructs a nested dictionary `trie` where each key is a character, and each value is a subtree. - The `insert_word` function inserts each word, creating new nodes as needed, and marks the end with a special character `'$'`. - The `print_trie` function recursively prints the trie in a tree-like structure, omitting null links. - The list `words` contains all keys to be inserted, as specified. - The trie is built by inserting each word, and then printed. --- **3. Output for the Given Code** When running this code, the output will be a visual representation of the trie structure, such as: ``` R-way trie structure: |-- a |-- i |-- d |-- c |-- o |-- m |-- e |-- f |-- o |-- r |-- g |-- o |-- o |-- d |-- i |-- s |-- n |-- o |-- w |-- o |-- f |-- p |-- e |-- o |-- p |-- l |-- e |-- t |-- h |-- e |-- e |-- t |-- o ``` **Steps to run the code:** 1. Save the code to a file named, for example, `trie.py`. 2. Ensure you have Python installed. 3. Run the script using the command: `python trie.py`. 4. Observe the printed trie structure in the console. This visualization helps understand how common prefixes are shared among words in the trie, illustrating the efficiency of prefix-based data retrieval.