Ternary Search Tree

A ternary search tree is a special trie data structure where the child nodes of a standard trie are ordered as a binary search tree.

Representation of ternary search trees:
Unlike trie(standard) data structure where each node contains 26 pointers for its children, each node in a ternary search tree contains only 3 pointers:
1. The left pointer points to the node whose value is less than the value in the current node.
2. The equal pointer points to the node whose value is equal to the value in the current node.
3. The right pointer points to the node whose value is greater than the value in the current node.

Apart from above three pointers, each node has a field to indicate data(character in case of dictionary) and another field to mark end of a string.
So, more or less it is similar to BST which stores data based on some order. However, data in a ternary search tree is distributed over the nodes. e.g. It needs 4 nodes to store the word “Geek”.
Below figure shows how exactly the words in a ternary search tree are stored?

One of the advantage of using ternary search trees over tries is that ternary search trees are a more space efficient (involve only three pointers per node as compared to 26 in standard tries). Further, ternary search trees can be used any time a hashtable would be used to store strings.

Tries are suitable when there is a proper distribution of words over the alphabets so that spaces are utilized most efficiently. Otherwise ternary search trees are better. Ternary search trees are efficient to use(in terms of space) when the strings to be stored share a common prefix.

Applications of ternary search trees:
1. Ternary search trees are efficient for queries like “Given a word, find the next word in dictionary(near-neighbor lookups)” or “Find all telephone numbers starting with 9342 or “typing few starting characters in a web browser displays all website names with this prefix”(Auto complete feature)”.

2. Used in spell checks: Ternary search trees can be used as a dictionary to store all the words. Once the word is typed in an editor, the word can be parallely searched in the ternary search tree to check for correct spelling.

Following is C implementation of ternary search tree. The operations implemented are, search, insert and traversal.

// C program to demonstrate Ternary Search Tree (TST) insert, travese 
// and search operations
#include <stdio.h>
#include <stdlib.h>
#define MAX 50

// A node of ternary search tree
struct Node
    char data;

    // True if this character is last character of one of the words
    unsigned isEndOfString: 1;

    struct Node *left, *eq, *right;

// A utility function to create a new ternary search tree node
struct Node* newNode(char data)
    struct Node* temp = (struct Node*) malloc(sizeof( struct Node ));
    temp->data = data;
    temp->isEndOfString = 0;
    temp->left = temp->eq = temp->right = NULL;
    return temp;

// Function to insert a new word in a Ternary Search Tree
void insert(struct Node** root, char *word)
    // Base Case: Tree is empty
    if (!(*root))
        *root = newNode(*word);

    // If current character of word is smaller than root's character,
    // then insert this word in left subtree of root
    if ((*word) < (*root)->data)
        insert(&( (*root)->left ), word);

    // If current character of word is greate than root's character,
    // then insert this word in right subtree of root
    else if ((*word) > (*root)->data)
        insert(&( (*root)->right ), word);

    // If current character of word is same as root's character,
        if (*(word+1))
            insert(&( (*root)->eq ), word+1);

        // the last character of the word
            (*root)->isEndOfString = 1;

// A recursive function to traverse Ternary Search Tree
void traverseTSTUtil(struct Node* root, char* buffer, int depth)
    if (root)
        // First traverse the left subtree
        traverseTSTUtil(root->left, buffer, depth);

        // Store the character of this node
        buffer[depth] = root->data;
        if (root->isEndOfString)
            buffer[depth+1] = '\0';
            printf( "%s\n", buffer);

        // Traverse the subtree using equal pointer (middle subtree)
        traverseTSTUtil(root->eq, buffer, depth + 1);

        // Finally Traverse the right subtree
        traverseTSTUtil(root->right, buffer, depth);

// The main function to traverse a Ternary Search Tree.
// It mainly uses traverseTSTUtil()
void traverseTST(struct Node* root)
    char buffer[MAX];
    traverseTSTUtil(root, buffer, 0);

// Function to search a given word in TST
int searchTST(struct Node *root, char *word)
    if (!root)
        return 0;

    if (*word < (root)->data)
        return searchTST(root->left, word);

    else if (*word > (root)->data)
        return searchTST(root->right, word);

        if (*(word+1) == '\0')
            return root->isEndOfString;

        return searchTST(root->eq, word+1);

// Driver program to test above functions
int main()
    struct Node *root = NULL;

    insert(&root, "cat");
    insert(&root, "cats");
    insert(&root, "up");
    insert(&root, "bug");

    printf("Following is traversal of ternary search tree\n");

    printf("\nFollowing are search results for cats, bu and cat respectively\n");
    searchTST(root, "cats")? printf("Found\n"): printf("Not Found\n");
    searchTST(root, "bu")?   printf("Found\n"): printf("Not Found\n");
    searchTST(root, "cat")?  printf("Found\n"): printf("Not Found\n");

    return 0;


Following is traversal of ternary search tree

Following are search results for cats, bu and cat respectively
Not Found

Time Complexity: The time complexity of the ternary search tree operations is similar to that of binary search tree. i.e. the insertion, deletion and search operations take time proportional to the height of the ternary search tree. The space is proportional to the length of the string to be stored.


This article is compiled by Aashish Barnwal and reviewed by GeeksforGeeks team. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

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  • 1234

    please explain me the traverse function.suppose string stored in cat,the a will be on left off c and after traversal string would become act,please explaim m confused

  • yzzzd

    could you share how to delete node in ternary search tree?

  • cooldude

    Java implementation

  • Amit Bgl

    wow code πŸ˜€

  • atiqwhiz

    I tried on the input

    and search BOA… output was “NOT FOUND”.

    I tried it differently you can see on my blog.

    /* Paste your code here (You may delete these lines if not writing code) */
    • atiqwhiz

      sorry it’s working …..great work

  • bateesh

    Can anybody explain me the complexity for inserting and searching in TST?

  • bateesh

    In the Search function
    When we have reached the end of string and the current node is not leaf node,then it will call for root->eq with current word as NULL.I think you can modify it to return
    root->isEndOfString instead of 1.Here is the modified version

    //when word finishes,then return whether current node is leaf or not
    if (*(word+1) == ‘\0′
    return root->isEndOfString;

    return searchTST(root->eq, word+1);

    • GeeksforGeeks

      @bateesh: Thanks for your inputs. We have updated the code. Keep it up!

  • lizard

    In the insert function you are inserting in the equal subtree when the node value is equal to the corresponding value of word….
    but in the figure shown above you have shown
    Please clarify….I have a bit confusion in this part.

    • http://www.facebook.com/barnwal.aashish Aashish

      Lets say we want to insert word CAT in the TST. We match the character C with the root node data(C), then deepen into the the equal pointer to insert the remaining suffix AT.

  • abhishek08aug

    Intelligent πŸ˜€

  • code_ignitor

    When the word “CUP” is searched the search returns 1 though the word was inserted.. is this correct?? Do help me …

    /* Paste your code here (You may delete these lines if not writing code) */
    • http://www.facebook.com/barnwal.aashish Aashish

      Please take a closer look at the search part. When the word “CUP” is searched, it will return false.
      Its because, when the character ‘C’ is referenced, it will deepen into the equal pointer where the subString “UP” will be searched. Observe the subTree with root node ‘A’. Searching “UP” onwards return false.

      • code_ignitor

        yeah Got it :) Great Thank you πŸ˜€

  • apsc

    Keep up the good work. Thanks for this. Please keep them coming especially tries, suffix trees, suffix array etc.

  • anandhakumar.P

    great explanation man . Thanks for it .

    can u answer this question

    “which data structure is the best for implementing telephone directory ” –> with time complexity

    • http://www.facebook.com/barnwal.aashish Aashish

      If time is at its premium, TRIE is more efficient. The time complexity will be proportional to the length of telephone number to be searched.

  • anonymous

    What happens when you add BBC after you add the above words ?

    • anonymous

      Works perfectly… Carefully see the traversal

  • monika

    Nice article.
    Can u please specify the applications where ternary search tree is better than Trie data-structure and vice-versa ?

    • http://www.facebook.com/barnwal.aashish Aashish

      Ternary search tree is applicable to all those applications where TRIE data structure is applicable. The choice depends on the type of operation and the density of data(determines space and time) i.e. how much the data is distributed over the alphabets(can be digits or other characters). The rule of thumb is:
      If time is at its premium, go for TRIE.
      If space is at its premium, go for Ternary search tree.

  • http://www.realinstagramfollowers.biz Riya Chowdhury

    wow! really nice way to make things clear. good job.

  • http:/theinternetbuzzer.com Ravi

    Nice Approach. Are you posting Everything about Data Structures?

    • Aashish

      We are posting important aspects of data structures.

  • Rahul

    can you give the diagram, how the tree is created after every string added ?

    • Aashish

      I encourage you to draw the tree yourself on paper. Please note that the tree formed differs if the order in which words are inserted in changed. e.g. Try to generate the tree with BUG, CATS, UP and CAT.