Are there any ways I can optimize the runtime of this function?

The assignment I'm working on takes a Binary Search tree of football teams and basically places them in sequence depending on the results of a tournament where each matchup is determined by an external function (did_x_beat_y). My IDE is telling me that I'm running into a SIGPIPE error and that my function is hanging for too long at times. Is there any way I can optimize this to improve the runtime?

// Return a BST containing a valid ordering of n teams
Node *order_n_teams(int n)
{
    
    Node *T = nullptr;
    
    // start by inserting the first team
    T = insert_random(T, 0, 0);
    
    // now insert the other teams...
    for (int i=1; i<n; i++) 
    {
        // insert team i so the sequence encoded by the BST remains valid
        if (did_x_beat_y(i, select(T,0)->key)) // can we insert at beginning?
        {
            T = insert_random(T, i, 0);
        }
        else if (did_x_beat_y(select(T,T->size-1)->key, i)) // can we insert at end?
        {
            T = insert_random(T, i, T->size);
        }
        
        //if team i doesn't go in beginning or end
        else 
        {
            int start = 0;
            
            int last = T->size - 1;
            
            while (start + 1 != last) 
            {
                
                int mid = (start + last) / 2;
                
                if (did_x_beat_y( i, select( T, mid)->key) )
                {
                    last = mid;
                }
                else
                {
                    start = mid;
                }
            }
            T = insert(T, i, last);
        }
    }
    
    return T;
}