Tiger Woods is 33 and already has more PGA Tour victories than Arnold Palmer. One of these days, he might have the kind of influence on players that the King can wield with a mere glint in his eye.

Consider the case of one player, whom every tournament would love to have in its field. He had no intention of playing the Arnold Palmer Invitational this year, for no other reason than he wanted to change up his schedule leading to the Masters. The young star was at Seminole in early March when he ran into Palmer, who greeted him with a smile, a firm handshake and a question.

“Are you coming to my tournament?” Palmer asked him.

Strangely enough, his plans changed that moment.

As far as anyone can tell, Woods never had that kind of moment with any of the big names who were somewhere other than his AT&T National last week. Only four of the top 12 players showed up at Congressional.

Would it have helped if Woods had done some recruiting?

“I’ve never asked anybody to play,” Woods said. “I don’t … I feel awkward doing that. If the guys can find time in their schedule to play, we’d love to have them.”

Congressional managed just fine without them.

The golf course, which will host its third U.S. Open in two years, is a classic that rivals any on the PGA Tour.

The tournament honors the military over the Fourth of July holiday outside the nation’s capital. Kids not only get in free, one marshal named Ron Fitzsimmons plucked a dozen of them out of the gallery each day and parked them behind the seventh tee so they could see. Proceeds will go toward building another Tiger Woods Learning Center, this one in the Washington area.

In three years, the AT&T National has become a model event.

Along with those trappings, it was quite a show inside the ropes. Woods played in the final pairing with the dynamic defending champion, Anthony Kim. He wound up beating another rising star, Hunter Mahan, who threw down a 62 in a final round that some 40,000 fans didn’t want to see end.

To no one’s surprise, the overnight TV rating was up 200 percent from last year.

“You’ve got Tiger and Anthony Kim in the last group,” Fred Couples said. “Does it really matter who’s not here?”

Even so, it was clear Couples was perturbed by the no-shows, and he wasn’t alone. There was plenty of grumbling going on outside the Beltway as tournament organizers quietly took roll.

More than 100 players earned over $1 million last year, and most of them owe that to Woods.

Was it too much to return the favor by playing his tournament?

None of the no-shows would have sold any more tickets. The crowd was estimated at over 160,000 for the week, and that doesn’t include Wednesday’s pro-am when Woods played with Dallas Cowboys quarterback Tony Romo in Redskins territory.

In a twist of fate, perhaps it was only fitting so much attention was paid to absentees.

Strength of field has been a debate since the days of Palmer and Jack Nicklaus, into the era of Couples and Greg Norman, yet no one has brought more attention to it than Woods.

Ask tournament directors if they would rather have nine of the top 10 players in the world without Woods, or only one player in the top 10 as long as that one was Woods. If they take Door No. 1, it’s only because they don’t have Tiger.

The perception is that if Woods plays, it’s a great field, even when it really isn’t. And when he doesn’t play, golf is irrelevant.

Four PGA Tour events that Woods missed this year had a higher-rated field than he attracted to the AT&T National, yet there was a story every week, in Los Angeles and Houston, in Fort Worth and Phoenix, about why Woods wasn’t there.

Remember, the only reason the AT&T National exists is because Jack Vickers folded his tent at Castle Pines after spending a decade trying to lure back Woods to the International (ignoring that he had Phil Mickelson, Ernie Els, Vijay Singh and Sergio Garcia every year).

The AT&T National has everything a player could want, much like Quail Hollow and the Memorial.

So why don’t more top players show up?

A strong field starts with a great course and a good spot on the schedule. What once appeared to be a great date — a week involving the Fourth of July — is starting to look suspect.

Sure, the Western Open rarely suffered when Woods, Nick Price and Tom Watson were winning against a strong field. However, that was when the PGA Tour season ended in early November, not the middle of September in the FedEx Cup configuration. A contracted schedule means more choices, and good events get crammed together in the summer.

Plus, more Europeans are in the top 50, and this is when their schedule actually takes them to Europe — France last week, the Scottish Open this week, then the British Open.

Don’t be surprised to see the AT&T National move to May if a spot in the schedule opens up in 2012, if not sooner.

In the meantime, Woods said all the right things Sunday night.

“For the guys who have come here over the last three years and played,” he said, “hopefully they’ve enjoyed their time.”

That’s all he can ask.

And as long as Woods is host of the tournament, that’s all it needs.