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# Suffix Tree Application 4 – Build Linear Time Suffix Array

Given a string, build it’s Suffix Array
We have already discussed following two ways of building suffix array:

Please go through these to have the basic understanding.
Here we will see how to build suffix array in linear time using suffix tree.
As a prerequisite, we must know how to build a suffix tree in one or the other way.
Here we will build suffix tree using Ukkonen’s Algorithm, discussed already as below:
Ukkonen’s Suffix Tree Construction – Part 1
Ukkonen’s Suffix Tree Construction – Part 2
Ukkonen’s Suffix Tree Construction – Part 3
Ukkonen’s Suffix Tree Construction – Part 4
Ukkonen’s Suffix Tree Construction – Part 5
Ukkonen’s Suffix Tree Construction – Part 6
Lets consider string abcabxabcd.
It’s suffix array would be:

`0 6 3 1 7 4 2 8 9 5`

Lets look at following figure:

If we do a DFS traversal, visiting edges in lexicographic order (we have been doing the same traversal in other Suffix Tree Application articles as well) and print suffix indices on leaves, we will get following:

`10 0 6 3 1 7 4 2 8 9 5`

“\$” is lexicographically lesser than [a-zA-Z].
The suffix index 10 corresponds to edge with “\$” label.
Except this 1st suffix index, the sequence of all other numbers gives the suffix array of the string.
So if we have a suffix tree of the string, then to get it’s suffix array, we just need to do a lexicographic order DFS traversal and store all the suffix indices in resultant suffix array, except the very 1st suffix index.

## C

 `// A C program to implement Ukkonen's Suffix Tree Construction``//  and then create suffix array in linear time``#include ``#include ``#include ``#define MAX_CHAR 256`` ` `struct` `SuffixTreeNode {``    ``struct` `SuffixTreeNode *children[MAX_CHAR];`` ` `    ``//pointer to other node via suffix link``    ``struct` `SuffixTreeNode *suffixLink;`` ` `    ``/*(start, end) interval specifies the edge, by which the``     ``node is connected to its parent node. Each edge will``     ``connect two nodes,  one parent and one child, and``     ``(start, end) interval of a given edge  will be stored``     ``in the child node. Lets say there are two nods A and B``     ``connected by an edge with indices (5, 8) then this``     ``indices (5, 8) will be stored in node B. */``    ``int` `start;``    ``int` `*end;`` ` `    ``/*for leaf nodes, it stores the index of suffix for``      ``the path  from root to leaf*/``    ``int` `suffixIndex;``};`` ` `typedef` `struct` `SuffixTreeNode Node;`` ` `char` `text[100]; ``//Input string``Node *root = NULL; ``//Pointer to root node`` ` `/*lastNewNode will point to newly created internal node,``  ``waiting for it's suffix link to be set, which might get``  ``a new suffix link (other than root) in next extension of``  ``same phase. lastNewNode will be set to NULL when last``  ``newly created internal node (if there is any) got it's``  ``suffix link reset to new internal node created in next``  ``extension of same phase. */``Node *lastNewNode = NULL;``Node *activeNode = NULL;`` ` `/*activeEdge is represented as input string character``  ``index (not the character itself)*/``int` `activeEdge = -1;``int` `activeLength = 0;`` ` `// remainingSuffixCount tells how many suffixes yet to``// be added in tree``int` `remainingSuffixCount = 0;``int` `leafEnd = -1;``int` `*rootEnd = NULL;``int` `*splitEnd = NULL;``int` `size = -1; ``//Length of input string`` ` `Node *newNode(``int` `start, ``int` `*end)``{``    ``Node *node =(Node*) ``malloc``(``sizeof``(Node));``    ``int` `i;``    ``for` `(i = 0; i < MAX_CHAR; i++)``          ``node->children[i] = NULL;`` ` `    ``/*For root node, suffixLink will be set to NULL``    ``For internal nodes, suffixLink will be set to root``    ``by default in  current extension and may change in``    ``next extension*/``    ``node->suffixLink = root;``    ``node->start = start;``    ``node->end = end;`` ` `    ``/*suffixIndex will be set to -1 by default and``      ``actual suffix index will be set later for leaves``      ``at the end of all phases*/``    ``node->suffixIndex = -1;``    ``return` `node;``}`` ` `int` `edgeLength(Node *n) {``    ``if``(n == root)``        ``return` `0;``    ``return` `*(n->end) - (n->start) + 1;``}`` ` `int` `walkDown(Node *currNode)``{``    ``/*activePoint change for walk down (APCFWD) using``     ``Skip/Count Trick  (Trick 1). If activeLength is greater``     ``than current edge length, set next  internal node as``     ``activeNode and adjust activeEdge and activeLength``     ``accordingly to represent same activePoint*/``    ``if` `(activeLength >= edgeLength(currNode))``    ``{``        ``activeEdge += edgeLength(currNode);``        ``activeLength -= edgeLength(currNode);``        ``activeNode = currNode;``        ``return` `1;``    ``}``    ``return` `0;``}`` ` `void` `extendSuffixTree(``int` `pos)``{``    ``/*Extension Rule 1, this takes care of extending all``    ``leaves created so far in tree*/``    ``leafEnd = pos;`` ` `    ``/*Increment remainingSuffixCount indicating that a``    ``new suffix added to the list of suffixes yet to be``    ``added in tree*/``    ``remainingSuffixCount++;`` ` `    ``/*set lastNewNode to NULL while starting a new phase,``     ``indicating there is no internal node waiting for``     ``it's suffix link reset in current phase*/``    ``lastNewNode = NULL;`` ` `    ``//Add all suffixes (yet to be added) one by one in tree``    ``while``(remainingSuffixCount > 0) {`` ` `        ``if` `(activeLength == 0)``            ``activeEdge = pos; ``//APCFALZ`` ` `        ``// There is no outgoing edge starting with``        ``// activeEdge from activeNode``        ``if` `(activeNode->children] == NULL)``        ``{``            ``//Extension Rule 2 (A new leaf edge gets created)``            ``activeNode->children] =``                                          ``newNode(pos, &leafEnd);`` ` `            ``/*A new leaf edge is created in above line starting``             ``from  an existing node (the current activeNode), and``             ``if there is any internal node waiting for it's suffix``             ``link get reset, point the suffix link from that last``             ``internal node to current activeNode. Then set lastNewNode``             ``to NULL indicating no more node waiting for suffix link``             ``reset.*/``            ``if` `(lastNewNode != NULL)``            ``{``                ``lastNewNode->suffixLink = activeNode;``                ``lastNewNode = NULL;``            ``}``        ``}``        ``// There is an outgoing edge starting with activeEdge``        ``// from activeNode``        ``else``        ``{``            ``// Get the next node at the end of edge starting``            ``// with activeEdge``            ``Node *next = activeNode->children];``            ``if` `(walkDown(next))``//Do walkdown``            ``{``                ``//Start from next node (the new activeNode)``                ``continue``;``            ``}``            ``/*Extension Rule 3 (current character being processed``              ``is already on the edge)*/``            ``if` `(text[next->start + activeLength] == text[pos])``            ``{``                ``//If a newly created node waiting for it's``                ``//suffix link to be set, then set suffix link``                ``//of that waiting node to current active node``                ``if``(lastNewNode != NULL && activeNode != root)``                ``{``                    ``lastNewNode->suffixLink = activeNode;``                    ``lastNewNode = NULL;``                ``}` `                ``//APCFER3``                ``activeLength++;``                ``/*STOP all further processing in this phase``                ``and move on to next phase*/``                ``break``;``            ``}`` ` `            ``/*We will be here when activePoint is in middle of``              ``the edge being traversed and current character``              ``being processed is not  on the edge (we fall off``              ``the tree). In this case, we add a new internal node``              ``and a new leaf edge going out of that new node. This``              ``is Extension Rule 2, where a new leaf edge and a new``            ``internal node get created*/``            ``splitEnd = (``int``*) ``malloc``(``sizeof``(``int``));``            ``*splitEnd = next->start + activeLength - 1;`` ` `            ``//New internal node``            ``Node *split = newNode(next->start, splitEnd);``            ``activeNode->children] = split;`` ` `            ``//New leaf coming out of new internal node``            ``split->children] = newNode(pos, &leafEnd);``            ``next->start += activeLength;``            ``split->children] = next;`` ` `            ``/*We got a new internal node here. If there is any``              ``internal node created in last extensions of same``              ``phase which is still waiting for it's suffix link``              ``reset, do it now.*/``            ``if` `(lastNewNode != NULL)``            ``{``            ``/*suffixLink of lastNewNode points to current newly``              ``created internal node*/``                ``lastNewNode->suffixLink = split;``            ``}`` ` `            ``/*Make the current newly created internal node waiting``              ``for it's suffix link reset (which is pointing to root``              ``at present). If we come across any other internal node``              ``(existing or newly created) in next extension of same``              ``phase, when a new leaf edge gets added (i.e. when``              ``Extension Rule 2 applies is any of the next extension``              ``of same phase) at that point, suffixLink of this node``              ``will point to that internal node.*/``            ``lastNewNode = split;``        ``}`` ` `        ``/* One suffix got added in tree, decrement the count of``          ``suffixes yet to be added.*/``        ``remainingSuffixCount--;``        ``if` `(activeNode == root && activeLength > 0) ``//APCFER2C1``        ``{``            ``activeLength--;``            ``activeEdge = pos - remainingSuffixCount + 1;``        ``}``        ``else` `if` `(activeNode != root) ``//APCFER2C2``        ``{``            ``activeNode = activeNode->suffixLink;``        ``}``    ``}``}`` ` `void` `print(``int` `i, ``int` `j)``{``    ``int` `k;``    ``for` `(k=i; k<=j; k++)``        ``printf``(``"%c"``, text[k]);``}`` ` `//Print the suffix tree as well along with setting suffix index``//So tree will be printed in DFS manner``//Each edge along with it's suffix index will be printed``void` `setSuffixIndexByDFS(Node *n, ``int` `labelHeight)``{``    ``if` `(n == NULL)  ``return``;`` ` `    ``if` `(n->start != -1) ``//A non-root node``    ``{``        ``//Print the label on edge from parent to current node``        ``//Uncomment below line to print suffix tree``       ``// print(n->start, *(n->end));``    ``}``    ``int` `leaf = 1;``    ``int` `i;``    ``for` `(i = 0; i < MAX_CHAR; i++)``    ``{``        ``if` `(n->children[i] != NULL)``        ``{``            ``//Uncomment below two lines to print suffix index``           ``// if (leaf == 1 && n->start != -1)``             ``//   printf(" [%d]\n", n->suffixIndex);`` ` `            ``//Current node is not a leaf as it has outgoing``            ``//edges from it.``            ``leaf = 0;``            ``setSuffixIndexByDFS(n->children[i], labelHeight +``                                  ``edgeLength(n->children[i]));``        ``}``    ``}``    ``if` `(leaf == 1)``    ``{``        ``n->suffixIndex = size - labelHeight;``        ``//Uncomment below line to print suffix index``        ``//printf(" [%d]\n", n->suffixIndex);``    ``}``}`` ` `void` `freeSuffixTreeByPostOrder(Node *n)``{``    ``if` `(n == NULL)``        ``return``;``    ``int` `i;``    ``for` `(i = 0; i < MAX_CHAR; i++)``    ``{``        ``if` `(n->children[i] != NULL)``        ``{``            ``freeSuffixTreeByPostOrder(n->children[i]);``        ``}``    ``}``    ``if` `(n->suffixIndex == -1)``        ``free``(n->end);``    ``free``(n);``}`` ` `/*Build the suffix tree and print the edge labels along with``suffixIndex. suffixIndex for leaf edges will be >= 0 and``for non-leaf edges will be -1*/``void` `buildSuffixTree()``{``    ``size = ``strlen``(text);``    ``int` `i;``    ``rootEnd = (``int``*) ``malloc``(``sizeof``(``int``));``    ``*rootEnd = - 1;`` ` `    ``/*Root is a special node with start and end indices as -1,``    ``as it has no parent from where an edge comes to root*/``    ``root = newNode(-1, rootEnd);`` ` `    ``activeNode = root; ``//First activeNode will be root``    ``for` `(i=0; isuffixIndex == -1) ``//If it is internal node``    ``{``        ``for` `(i = 0; i < MAX_CHAR; i++)``        ``{``            ``if``(n->children[i] != NULL)``            ``{``                ``doTraversal(n->children[i], suffixArray, idx);``            ``}``        ``}``    ``}``    ``//If it is Leaf node other than "\$" label``    ``else` `if``(n->suffixIndex > -1 && n->suffixIndex < size)``    ``{``        ``suffixArray[(*idx)++] = n->suffixIndex;``    ``}``}` `void` `buildSuffixArray(``int` `suffixArray[])``{``    ``int` `i = 0;``    ``for``(i=0; i< size; i++)``        ``suffixArray[i] = -1;``    ``int` `idx = 0;``    ``doTraversal(root, suffixArray, &idx);``    ``printf``(``"Suffix Array for String "``);``    ``for``(i=0; i

## C++

 `// A CPP program to implement Ukkonen's Suffix Tree Construction``//  and then create suffix array in linear time``#include``using` `namespace` `std;``#define MAX_CHAR 256`` ` `struct` `SuffixTreeNode {``    ``struct` `SuffixTreeNode *children[MAX_CHAR];`` ` `    ``//pointer to other node via suffix link``    ``struct` `SuffixTreeNode *suffixLink;`` ` `    ``/*(start, end) interval specifies the edge, by which the``     ``node is connected to its parent node. Each edge will``     ``connect two nodes,  one parent and one child, and``     ``(start, end) interval of a given edge  will be stored``     ``in the child node. Lets say there are two nods A and B``     ``connected by an edge with indices (5, 8) then this``     ``indices (5, 8) will be stored in node B. */``    ``int` `start;``    ``int` `*end;`` ` `    ``/*for leaf nodes, it stores the index of suffix for``      ``the path  from root to leaf*/``    ``int` `suffixIndex;``};`` ` `typedef` `struct` `SuffixTreeNode Node;`` ` `char` `text[100]; ``//Input string``Node *root = NULL; ``//Pointer to root node`` ` `/*lastNewNode will point to newly created internal node,``  ``waiting for it's suffix link to be set, which might get``  ``a new suffix link (other than root) in next extension of``  ``same phase. lastNewNode will be set to NULL when last``  ``newly created internal node (if there is any) got it's``  ``suffix link reset to new internal node created in next``  ``extension of same phase. */``Node *lastNewNode = NULL;``Node *activeNode = NULL;`` ` `/*activeEdge is represented as input string character``  ``index (not the character itself)*/``int` `activeEdge = -1;``int` `activeLength = 0;`` ` `// remainingSuffixCount tells how many suffixes yet to``// be added in tree``int` `remainingSuffixCount = 0;``int` `leafEnd = -1;``int` `*rootEnd = NULL;``int` `*splitEnd = NULL;``int` `size = -1; ``//Length of input string`` ` `Node *newNode(``int` `start, ``int` `*end)``{``    ``Node *node =(Node*) ``malloc``(``sizeof``(Node));``    ``int` `i;``    ``for` `(i = 0; i < MAX_CHAR; i++)``          ``node->children[i] = NULL;`` ` `    ``/*For root node, suffixLink will be set to NULL``    ``For internal nodes, suffixLink will be set to root``    ``by default in  current extension and may change in``    ``next extension*/``    ``node->suffixLink = root;``    ``node->start = start;``    ``node->end = end;`` ` `    ``/*suffixIndex will be set to -1 by default and``      ``actual suffix index will be set later for leaves``      ``at the end of all phases*/``    ``node->suffixIndex = -1;``    ``return` `node;``}`` ` `int` `edgeLength(Node *n) {``    ``if``(n == root)``        ``return` `0;``    ``return` `*(n->end) - (n->start) + 1;``}`` ` `int` `walkDown(Node *currNode)``{``    ``/*activePoint change for walk down (APCFWD) using``     ``Skip/Count Trick  (Trick 1). If activeLength is greater``     ``than current edge length, set next  internal node as``     ``activeNode and adjust activeEdge and activeLength``     ``accordingly to represent same activePoint*/``    ``if` `(activeLength >= edgeLength(currNode))``    ``{``        ``activeEdge += edgeLength(currNode);``        ``activeLength -= edgeLength(currNode);``        ``activeNode = currNode;``        ``return` `1;``    ``}``    ``return` `0;``}`` ` `void` `extendSuffixTree(``int` `pos)``{``    ``/*Extension Rule 1, this takes care of extending all``    ``leaves created so far in tree*/``    ``leafEnd = pos;`` ` `    ``/*Increment remainingSuffixCount indicating that a``    ``new suffix added to the list of suffixes yet to be``    ``added in tree*/``    ``remainingSuffixCount++;`` ` `    ``/*set lastNewNode to NULL while starting a new phase,``     ``indicating there is no internal node waiting for``     ``it's suffix link reset in current phase*/``    ``lastNewNode = NULL;`` ` `    ``//Add all suffixes (yet to be added) one by one in tree``    ``while``(remainingSuffixCount > 0) {`` ` `        ``if` `(activeLength == 0)``            ``activeEdge = pos; ``//APCFALZ`` ` `        ``// There is no outgoing edge starting with``        ``// activeEdge from activeNode``        ``if` `(activeNode->children] == NULL)``        ``{``            ``//Extension Rule 2 (A new leaf edge gets created)``            ``activeNode->children] =``                                          ``newNode(pos, &leafEnd);`` ` `            ``/*A new leaf edge is created in above line starting``             ``from  an existing node (the current activeNode), and``             ``if there is any internal node waiting for it's suffix``             ``link get reset, point the suffix link from that last``             ``internal node to current activeNode. Then set lastNewNode``             ``to NULL indicating no more node waiting for suffix link``             ``reset.*/``            ``if` `(lastNewNode != NULL)``            ``{``                ``lastNewNode->suffixLink = activeNode;``                ``lastNewNode = NULL;``            ``}``        ``}``        ``// There is an outgoing edge starting with activeEdge``        ``// from activeNode``        ``else``        ``{``            ``// Get the next node at the end of edge starting``            ``// with activeEdge``            ``Node *next = activeNode->children];``            ``if` `(walkDown(next))``//Do walkdown``            ``{``                ``//Start from next node (the new activeNode)``                ``continue``;``            ``}``            ``/*Extension Rule 3 (current character being processed``              ``is already on the edge)*/``            ``if` `(text[next->start + activeLength] == text[pos])``            ``{``                ``//If a newly created node waiting for it's``                ``//suffix link to be set, then set suffix link``                ``//of that waiting node to current active node``                ``if``(lastNewNode != NULL && activeNode != root)``                ``{``                    ``lastNewNode->suffixLink = activeNode;``                    ``lastNewNode = NULL;``                ``}` `                ``//APCFER3``                ``activeLength++;``                ``/*STOP all further processing in this phase``                ``and move on to next phase*/``                ``break``;``            ``}`` ` `            ``/*We will be here when activePoint is in middle of``              ``the edge being traversed and current character``              ``being processed is not  on the edge (we fall off``              ``the tree). In this case, we add a new internal node``              ``and a new leaf edge going out of that new node. This``              ``is Extension Rule 2, where a new leaf edge and a new``            ``internal node get created*/``            ``splitEnd = (``int``*) ``malloc``(``sizeof``(``int``));``            ``*splitEnd = next->start + activeLength - 1;`` ` `            ``//New internal node``            ``Node *split = newNode(next->start, splitEnd);``            ``activeNode->children] = split;`` ` `            ``//New leaf coming out of new internal node``            ``split->children] = newNode(pos, &leafEnd);``            ``next->start += activeLength;``            ``split->children] = next;`` ` `            ``/*We got a new internal node here. If there is any``              ``internal node created in last extensions of same``              ``phase which is still waiting for it's suffix link``              ``reset, do it now.*/``            ``if` `(lastNewNode != NULL)``            ``{``            ``/*suffixLink of lastNewNode points to current newly``              ``created internal node*/``                ``lastNewNode->suffixLink = split;``            ``}`` ` `            ``/*Make the current newly created internal node waiting``              ``for it's suffix link reset (which is pointing to root``              ``at present). If we come across any other internal node``              ``(existing or newly created) in next extension of same``              ``phase, when a new leaf edge gets added (i.e. when``              ``Extension Rule 2 applies is any of the next extension``              ``of same phase) at that point, suffixLink of this node``              ``will point to that internal node.*/``            ``lastNewNode = split;``        ``}`` ` `        ``/* One suffix got added in tree, decrement the count of``          ``suffixes yet to be added.*/``        ``remainingSuffixCount--;``        ``if` `(activeNode == root && activeLength > 0) ``//APCFER2C1``        ``{``            ``activeLength--;``            ``activeEdge = pos - remainingSuffixCount + 1;``        ``}``        ``else` `if` `(activeNode != root) ``//APCFER2C2``        ``{``            ``activeNode = activeNode->suffixLink;``        ``}``    ``}``}`` ` `void` `print(``int` `i, ``int` `j)``{``    ``int` `k;``    ``for` `(k=i; k<=j; k++)``        ``cout<start != -1) ``//A non-root node``    ``{``        ``//Print the label on edge from parent to current node``        ``//Uncomment below line to print suffix tree``       ``// print(n->start, *(n->end));``    ``}``    ``int` `leaf = 1;``    ``int` `i;``    ``for` `(i = 0; i < MAX_CHAR; i++)``    ``{``        ``if` `(n->children[i] != NULL)``        ``{``            ``//Uncomment below two lines to print suffix index``           ``// if (leaf == 1 && n->start != -1)``            ` ` ` `            ``//Current node is not a leaf as it has outgoing``            ``//edges from it.``            ``leaf = 0;``            ``setSuffixIndexByDFS(n->children[i], labelHeight +``                                  ``edgeLength(n->children[i]));``        ``}``    ``}``    ``if` `(leaf == 1)``    ``{``        ``n->suffixIndex = size - labelHeight;``        ``//Uncomment below line to print suffix index``       ` `    ``}``}`` ` `void` `freeSuffixTreeByPostOrder(Node *n)``{``    ``if` `(n == NULL)``        ``return``;``    ``int` `i;``    ``for` `(i = 0; i < MAX_CHAR; i++)``    ``{``        ``if` `(n->children[i] != NULL)``        ``{``            ``freeSuffixTreeByPostOrder(n->children[i]);``        ``}``    ``}``    ``if` `(n->suffixIndex == -1)``        ``free``(n->end);``    ``free``(n);``}`` ` `/*Build the suffix tree and print the edge labels along with``suffixIndex. suffixIndex for leaf edges will be >= 0 and``for non-leaf edges will be -1*/``void` `buildSuffixTree()``{``    ``size = ``strlen``(text);``    ``int` `i;``    ``rootEnd = (``int``*) ``malloc``(``sizeof``(``int``));``    ``*rootEnd = - 1;`` ` `    ``/*Root is a special node with start and end indices as -1,``    ``as it has no parent from where an edge comes to root*/``    ``root = newNode(-1, rootEnd);`` ` `    ``activeNode = root; ``//First activeNode will be root``    ``for` `(i=0; isuffixIndex == -1) ``//If it is internal node``    ``{``        ``for` `(i = 0; i < MAX_CHAR; i++)``        ``{``            ``if``(n->children[i] != NULL)``            ``{``                ``doTraversal(n->children[i], suffixArray, idx);``            ``}``        ``}``    ``}``    ``//If it is Leaf node other than "\$" label``    ``else` `if``(n->suffixIndex > -1 && n->suffixIndex < size)``    ``{``        ``suffixArray[(*idx)++] = n->suffixIndex;``    ``}``}` `void` `buildSuffixArray(``int` `suffixArray[])``{``    ``int` `i = 0;``    ``for``(i=0; i< size; i++)``        ``suffixArray[i] = -1;``    ``int` `idx = 0;``    ``doTraversal(root, suffixArray, &idx);``    ``cout<<``"Suffix Array for String "``;``    ``for``(i=0; i

Output:

```Suffix Array for String banana is: 5 3 1 0 4 2
Suffix Array for String GEEKSFORGEEKS is: 9 1 10 2 5 8 0 11 3 6 7 12 4
Suffix Array for String AAAAAAAAAA is: 9 8 7 6 5 4 3 2 1 0
Suffix Array for String ABCDEFG is: 0 1 2 3 4 5 6
Suffix Array for String ABABABA is: 6 4 2 0 5 3 1
Suffix Array for String abcabxabcd is: 0 6 3 1 7 4 2 8 9 5
Suffix Array for String CCAAACCCGATTA is: 12 2 3 4 9 1 0 5 6 7 8 11 10```

Ukkonen’s Suffix Tree Construction takes O(N) time and space to build suffix tree for a string of length N and after that, traversal of tree take O(N) to build suffix array.
So overall, it’s linear in time and space.
Can you see why traversal is O(N) ?? Because a suffix tree of string of length N will have at most N-1 internal nodes and N leaves. Traversal of these nodes can be done in O(N).
We have published following more articles on suffix tree applications: