# Ternary search tree

Ternary Search Tree (TST)
Typetree
Time complexity in big O notation
Algorithm Search Average Worst case O(log n) O(n) O(log n) O(n) O(log n) O(n)

In computer science, a ternary search tree is a type of trie (sometimes called a prefix tree) where nodes are arranged in a manner similar to a binary search tree, but with up to three children rather than the binary tree's limit of two. Like other prefix trees, a ternary search tree can be used as an associative map structure with the ability for incremental string search. However, ternary search trees are more space efficient compared to standard prefix trees, at the cost of speed. Common applications for ternary search trees include spell-checking and auto-completion.

## Description

Each node of a ternary search tree stores a single character, an object (or a pointer to an object depending on implementation), and pointers to its three children conventionally named equal kid, lo kid and hi kid, which can also be referred respectively as middle (child), lower (child) and higher (child). A node may also have a pointer to its parent node as well as an indicator as to whether or not the node marks the end of a word. The lo kid pointer must point to a node whose character value is less than the current node. The hi kid pointer must point to a node whose character is greater than the current node. The equal kid points to the next character in the word. The figure below shows a ternary search tree with the strings "cute","cup","at","as","he","us" and "i":

```          c
/ | \
a  u  h
|  |  | \
t  t  e  u
/  / |   / |
s  p  e  i  s
```

As with other trie data structures, each node in a ternary search tree represents a prefix of the stored strings. All strings in the middle subtree of a node start with that prefix.

## Operations

### Insertion

Inserting a value into a ternary search can be defined recursively or iteratively much as lookups are defined. This recursive method is continually called on nodes of the tree given a key which gets progressively shorter by pruning characters off the front of the key. If this method reaches a node that has not been created, it creates the node and assigns it the character value of the first character in the key. Whether a new node is created or not, the method checks to see if the first character in the string is greater than or less than the character value in the node and makes a recursive call on the appropriate node as in the lookup operation. If, however, the key's first character is equal to the node's value then the insertion procedure is called on the equal kid and the key's first character is pruned away. Like binary search trees and other data structures, ternary search trees can become degenerate depending on the order of the keys.[self-published source?] Inserting keys in alphabetical order is one way to attain the worst possible degenerate tree. Inserting the keys in random order often produces a well-balanced tree.

```function insertion(string key) is
node p := root
//initialized to be equal in case root is null
node last := root
int idx := 0
while p is not null do
//recurse on proper subtree
if key[idx] < p.splitchar then
last := p
p := p.left
else if key[idx] > p.splitchar then
last := p
p := p.right
else:
// key is already in our Tree
if idx == length(key) then
return
//trim character from our key
idx := idx+1
last := p
p := p.mid
p := node()
//add p in as a child of the last non-null node (or root if root is null)
if root == null then
root := p
else if last.splitchar < key[idx] then
last.right := p
else if last.splitchar > key[idx] then
last.left := p
else
last.mid := p
p.splitchar := key[idx]
idx := idx+1
// Insert remainder of key
while idx < length(key) do
p.mid := node()
p.mid.splitchar := key[idx]
idx += 1
```

### Search

To look up a particular node or the data associated with a node, a string key is needed. A lookup procedure begins by checking the root node of the tree and determining which of the following conditions has occurred. If the first character of the string is less than the character in the root node, a recursive lookup can be called on the tree whose root is the lo kid of the current root. Similarly, if the first character is greater than the current node in the tree, then a recursive call can be made to the tree whose root is the hi kid of the current node. As a final case, if the first character of the string is equal to the character of the current node then the function returns the node if there are no more characters in the key. If there are more characters in the key then the first character of the key must be removed and a recursive call is made given the equal kid node and the modified key. This can also be written in a non-recursive way by using a pointer to the current node and a pointer to the current character of the key.

#### Pseudocode

``` function search(string query) is
if is_empty(query) then
return false

node p := root
int idx := 0

while p is not null do
if query[idx] < p.splitchar then
p := p.left
else if query[idx] > p.splitchar then
p := p.right;
else
if idx = length(query) then
return true
idx := idx + 1
p := p.mid

return false
```

### Deletion

The delete operation consists of searching for a key string in the search tree and finding a node, called firstMid in the below pseudocode, such that the path from the middle child of firstMid to the end of the search path for the key string has no left or right children. This would represent a unique suffix in the ternary tree corresponding to the key string. If there is no such path, this means that the key string is either fully contained as a prefix of another string, or is not in the search tree. Many implementations make use of an end of string character to ensure only the latter case occurs. The path is then deleted from firstMid.mid to the end of the search path. In the case that firstMid is the root, the key string must have been the last string in the tree, and thus the root is set to null after the deletion.

``` function delete(string key) is
if is_empty(key) then
return

node p := root
int idx := 0

node firstMid := null
while p is not null do
if key[idx] < p.splitchar then
firstMid := null
p := p.left
else if key[idx] > p.splitchar then
firstMid := null
p := p.right
else
firstMid := p
while p is not null and key[idx] == p.splitchar do
idx := idx + 1
p := p.mid

if firstMid == null then
return  // No unique string suffix

// At this point, firstMid points to the node before the strings unique suffix occurs
node q := firstMid.mid
node p := q
firstMid.mid := null // disconnect suffix from tree
while q is not null do //walk down suffix path and delete nodes
p := q
q := q.mid
delete(p)			// free memory associated with node p
if firstMid == root then
delete(root)       //delete the entire tree
root := null
```

## Running time

The running time of ternary search trees varies significantly with the input. Ternary search trees run best when given several similar strings, especially when those strings share a common prefix. Alternatively, ternary search trees are effective when storing a large number of relatively short strings (such as words in a dictionary). Running times for ternary search trees are similar to binary search trees, in that they typically run in logarithmic time, but can run in linear time in the degenerate (worst) case. Further, the size of the strings must also be kept in mind when considering runtime. For example, in the search path for a string of length k, there will be k traversals down middle children in the tree, as well as a logarithmic number of traversals down left and right children in the tree. Thus, in a ternary search tree on a small number of very large strings the lengths of the strings can dominate the runtime.

Time complexities for ternary search tree operations:

Average-case running time Worst-case running time
Lookup O(log n + k) O(n + k)
Insertion O(log n + k) O(n + k)
Delete O(log n + k) O(n + k)

## Comparison to other data structures

### Tries

While being slower than other prefix trees, ternary search trees can be better suited for larger data sets due to their space-efficiency.

### Hash maps

Hashtables can also be used in place of ternary search trees for mapping strings to values. However, hash maps also frequently use more memory than ternary search trees (but not as much as tries). Additionally, hash maps are typically slower at reporting a string that is not in the same data structure, because it must compare the entire string rather than just the first few characters. There is some evidence that shows ternary search trees running faster than hash maps. Additionally, hash maps do not allow for many of the uses of ternary search trees, such as near-neighbor lookups.

### DAFSAs (deterministic acyclic finite state automaton)

If storing dictionary words is all that is required (i.e., storage of information auxiliary to each word is not required), a minimal deterministic acyclic finite state automaton (DAFSA) would use less space than a trie or a ternary search tree. This is because a DAFSA can compress identical branches from the trie which correspond to the same suffixes (or parts) of different words being stored.

## Uses

Ternary search trees can be used to solve many problems in which a large number of strings must be stored and retrieved in an arbitrary order. Some of the most common or most useful of these are below: