I'm no expert on tryn, however, I strongly discourage you from staying in the jungle too long. After cleaning your own half of the jungle completely you should be level 4. At that time you have to decide whether to go for the enemy jungle or a gank.
The main problem is that a gank is most often the desired way to go. No matter that you don't have level 6 yet. You do have the benefit of surprise. In most games, at this time one of your lanes is in need of some help. If you stay in the enemy jungle, the flaming will soon reach you there for leaving your mates. Even if the game goes well, you often have an enemy jungler who will make life miserable for you if you encounter him in his own jungle (not by himself, but you're much closer to his mates, than your own).
tl;dr: it's a team game, so you shouldn't pretend to play it solo in a deserted jungle. Join the fun asap (you may no have ulti, but you should have auras+surprise).
Edit: As for the Smite-problem itself: Tryn is a very slow jungler even if you take smite along. I fear, that even if you could do without, you would lose up so much time, that you will again be incapable to keep up with others. (An alternative I have sometimes seen is for Tryn to join a lane in the beginning, then hit the jungle lateron at lvls3-5.)