(CNN) -- A few seconds could have made all the difference in the fate of a family who spent three days lost in the snowy California woods, according to the helicopter pilot who found them.



Josh and Lexi Dominguez exit a helicopter Wednesday, shortly after being found.



The father of the family, Frederick Dominguez, came running out of the culvert where they had sought shelter when family members heard the sound of the California Highway Patrol helicopter Wednesday afternoon.



"Had he not been moving, we would not have seen him, because the tree line was very dense and he came climbing out of the culvert," helicopter pilot Steve Ward told CNN on Thursday. "We were just very lucky."



Dominguez had arranged branches to spell the word "help" near the culvert, but rescuers didn't see that until they were turning the helicopter around after spotting Dominguez.



The helicopter was on its way out of the area at that point, trying to get ahead of bad weather.



"The small window of opportunity we had to find them, it was nothing short of a miracle," paramedic David White, who was riding with Ward, told CNN on Thursday.  Watch White and Ward describe the rescue »



Dominguez and his three children had been lost in the snow since Sunday, when they set out to cut down a Christmas tree.



They sought shelter first in a lean-to they made of branches, then in the culvert under a road.  See family's photos of their ordeal »



Dominguez said his daughter Lexi, 14, was the first to hear the helicopter overhead. He said he ran though several feet of snow barefooted to wave it down.



"When they turned around, man, I was just praising God and saying, 'Thank you, Lord. Thank you, Lord,' because I knew we had made it," he said.



Thursday, Lexi was hospitalized after complaining that her feet hurt. Trying to protect their cold, wet feet from frostbite was an ongoing problem while the four were lost in the forest.



Through their three-day ordeal, the oldest son, Chris, 18, tried to keep his younger brother and sister optimistic.



"I didn't want them to really lose hope," he told CNN's Anderson Cooper on Wednesday night, hours after the family was rescued. "Whenever they would freak out, I would just be like, it's all right. We're going to make it through this. This is nothing. Like, we have already been here a couple of days. What's a couple more days?"  Watch the family talk about their "scary" ordeal »



But it was hard to stay hopeful lost in rugged terrain, in heavy snow, with no food and few warm clothes.



"I just remember walking and walking and being like, we're not going to make it," said Lexi. "They can't even see us through all this fog. And I just -- there was just a couple of times where I was really, really scared."



"I didn't think we were going to make it," said Josh, 12.



They removed their sodden socks in an effort to stay warm and dry while they waited for rescue, according to the rescuers who found them. They warmed their feet inside each other's shirts to help stave off frostbite.



"You just go to survival mode," Frederick Dominguez said. "Every parent would do that. You would do anything, sacrifice yourself, because these are your kids."  Watch mother as she finds out family is safe »



While the family huddled in the culvert, Lexi led them in impromptu singalongs.



"Someone would say, sing this song, and I would be like, OK," she recalled.



"We would all help her, too," her brother Josh added.



Chris told CNN one of the things he would remember most about the experience was "Lexi in there, in the tunnel, singing her heart
Who was  hospitalized after complaining about her feet hurting? Who set out Sunday to cut the Christmas tree? Who was hospitalized after complaining her feet hurt? Where did a family seek shelter? What did Frederick Dominguez and his 3 children set out Sunday to cut? Who was hospitalized due to sore feet? What did Frederick Dominguez and his 3 children set out to do on Sunday? What did the family do in the shelter?
Lexi Dominguez and his three children Lexi culvert a Christmas tree. Lexi cut down a Christmas tree. They removed their sodden socks in an effort to stay warm and dry while they waited for rescue,
question: Who was  hospitalized after complaining about her feet hurting?, answer: Lexi | question: Who set out Sunday to cut the Christmas tree?, answer: Dominguez and his three children | question: Who was hospitalized after complaining her feet hurt?, answer: Lexi | question: Where did a family seek shelter?, answer: culvert | question: What did Frederick Dominguez and his 3 children set out Sunday to cut?, answer: a Christmas tree. | question: Who was hospitalized due to sore feet?, answer: Lexi | question: What did Frederick Dominguez and his 3 children set out to do on Sunday?, answer: cut down a Christmas tree. | question: What did the family do in the shelter?, answer: They removed their sodden socks in an effort to stay warm and dry while they waited for rescue,