# 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 isn’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(n2), but Esko Ukkonen discovered a way to construct a suffix tree in O(n) (linear time).