Longest repeated substring

Question

I am trying to solve a problem- Longest repeated substring in a string. Firstly, I built a suffix tree that takes O(n) time and then I traversed the suffix tree to find the deepest internal node. I am not sure whether traversing in a suffix tree would be O(n) or not?

These are the properties of a suffix tree-

1. A suffix tree will contain n leaves which are numbered from 1 to n.
2. Each internal node (except root) should have at least 2 children.

If each internal node can have more than 2 children,How can I say that number of nodes in a suffix tree is O(n)? So, How will the complexity be O(n)?

Here is the code that I have written for traversal. I am aware that this platform is language independent so I have added the relevant comments along with my code.

struct StNode
{
    string str;
    StNode* child[256];
};
void lrs(StNode* t,string s,string &res)
{
    if(leafNode(t)){    //leafNode means no child
        string str=t->str;
        s.erase(s.length()-str.length());   //erasing the leafNode's string
        int k=s.length();
        if(k>res.length())   res=s;
        return;
    }
    for(int i=0;i<256;i++){     //traversing through all childs
        if(t->child[i]!=NULL){
            string str=t->child[i]->str;
            lrs(t->child[i],s+str,res);
        }
    }
}