# Suffix Trees

25 Oct 2016# Suffix Trees

## What is a Suffix Tree?

A suffix tree is a data sructure that contains each suffix of a string, a suffix is defined as the end part of a string. For example, the suffixes for the string banana are:

- a
- na
- ana
- nana
- anana
- banana

Notice that the entire string is also considered a suffix.
We can also include the empty string as a suffix, so in order to include the empty string we need to append a character that isnâ€™t in the alphabet to the string.
We will assume that the character `$`

is not in the alphabet.
Our string is now banana$, and the suffixes are:

- $
- a$
- na$
- ana$
- nana$
- anana$
- banana$

## How to Construct a Suffix Tree

The easiest way to understand how to construct a suffix tree is to first construct a suffix trie, then collapse nodes to convert the trie to a tree. Here is the graphical representation of the suffix trie:

Now we collapse the nodes that only have one child, and the resulting suffix tree is this:

## How to Use a Suffix Tree

Observe that each suffix is represented in this structure.
This means that we can efficiently search for a suffix (or substring) by traversing the suffix tree.
For example if we were to search for the substring `ana`

in the suffix tree for `banana$`

, this would be the traversal path:

`0 -> 2 -> 3`

*Note:* We can stop at any node because we are searching for a substring rather than a suffix, if we were searching for a suffix, the search string would have been `ana$`

.

## Suffix Tree Construction

How would you create a naive algorithm to construct a suffix tree?
You can easily create a simple algorithm to construct a suffix tree in *O(n ^{2})*, but Esko Ukkonen discovered a way to construct a suffix tree in

*O(n)*(linear time).