This order addition of characters gives Ukkonen's algorithm its "on-line" property. The original algorithm presented by Peter Weiner proceeded backward from the last character to the first one from the shortest to the longest suffix. A simpler algorithm was found by Edward M. McCreight, going from the longest to the shortest suffix.
The addition of characters is what makes Ukkonen's algorithm "on-line." The original algorithm by Peter Weiner worked backward from the last character to the first one, prioritizing suffixes based on length. Edward M. McCreight created a simpler algorithm that starts with the longest suffix and moves towards the shortest.