Tree Buchet

In animation, seeing a tall, skinny tree anywhere is a sure sign that someone is about to go flying.

In the world of cartoons, any sufficiently tall, flexible tree can be transformed into a Tree Buchet. The formula is generally as follows:

Have the villain chase the hero through a forest, tree farm, or any other sufficiently wooded area.
In an attempt to escape the villain, have the hero run up the tallest, skinniest tree available.
Have the villain chase the hero up the tree. The fairly fragile tree will begin to bend beneath the weight of both people trapped within it, frequently doubling over.
The hero must now jump out of the tree.
We have a few seconds to register the look of terror on the villain's face before the tree, freed of the excess weight, snaps back up. Thwwwwpp! The villain goes flying.

A frequent variation is for a character to use a tree snare—a tree purposefully bent low, with a trap tied to one end. Usually done in an attempt to Catch That Pigeon. Usually, the hunter will find himself either riding the tree out of town, or thrown back and forth after they get caught in his own trap.

Another variation is the villain choosing to catapult himself with a tree, either to reach something higher or to jump over a wall. The effect varies from catapulting directly into the ground to missing the target by one inch.

Yet another variation is the hero using it to launch himself. This works, unless it's too early in the story for success.

Tree Buchets are named after trebuchets (pronounced 'treh-byu-shay'). Note that the classic Tree Buchet is not an actual trebuchet: trebuchets work using a lever with heavy counterweights. Tree Buchets work because trees are apparently perfectly elastic.

Compare Catapult to Glory, where this is done with an actual catapult, and Caught in a Snare, which to catch a man-sized target needs similarly elastic trees.

Examples of Tree Buchet include:

Advertising

There's a commercial right now for McDonald's that results in Ronald flying over the city with apples that drop randomly into the hands of children so they have healthy snacks.
Happens in a Pop Tarts commercial, too.

Anime & Manga

Done with a giant cactus and a cruise liner in Digimon Adventure episode 17.

Comics

Subverted in Carl Barks' 1953 Uncle Scrooge story "Back to the Klondike". A hungry bear chases Donald Duck up a tree; the tree bends down... but rather than Donald jumping off the tree, the bear falls off... and Donald is sent flying.
- Dick Kinney's 1965 Donald Duck story The Pioneer. An hungry bear chases Donald's cousin Fethry up a tree; the tree bends down and Fethry jumps off, sending the bear flying. Later, though, the bear gets even, chasing Fethry up another tree and beating him to jumping off.
One issue of the old Sgt. Rock comic has a soldier obsessed with Superman comics use a tree to launch himself over a gunner's nest so he can grenade it.

Films -- Animation

In the first Shrek movie, Shreck bends a tree to use as a bridge for Fiona. After she crosses, he lets the tree go... unwittingly (or maybe not) sending poor Donkey flying.
In Kung Fu Panda, Po tries to launch himself with a bamboo catapult, but he's just too heavy. When he gets off, the bamboo hits him in the face as it snaps back, then hits him again in the head on the rebound. He uses this trick to his advantage during his fight with Tai Lung at the climax of the film.
Horton gets rid of Vlad the vulture this way in The Movie of Horton Hears a Who. The elephant manages a Bond One-Liner before he does, then mentions that he usually doesn't come up with those until later.
The first two Ice Age movies do something with this. In one it uses a tilting rock to launch Sid the Sloth. In the other the classic tree. The tree in question is not only a full grown pine, but also being held down by Manny the Mammoth.
Happens to one of the two dogs that were constantly pursuing Edgar during their first chase scene from The Aristocats.

Films -- Live Action

The troll in the beginning of Enchanted has this happen to him.
Shows up in Crouching Tiger, Hidden Dragon, during the bamboo forest fight.
In Indiana Jones and the Kingdom of the Crystal Skull, an amphibian vehicle lands on a huge tree that was growing on the ravine's side. The car is gently lowered in to the river below, then the tree recoils and hits the cliff wall, knocking down several Soviet soldiers.
In Pirates of the Caribbean: On Stranger Tides, Jack launches himself from palm tree to palm tree to get across the Spaniards' encampment and escape.

Folklore

An Urban Legend version of this story turned up in the Darwin Awards. According to this tale, a heavy storm caused a tree to get bent over and wedged under the eaves of a house. The homeowner climbed up the tree to saw the top of the tree off to free it from the eaves. Once he cut through, the tree sprung back, catapulting the owner to his death (four miles away according to some versions).
There's an Urban Legend that goes something like this- An elderly woman is in a panic because her cat is stuck in a tree. Her neighbor notices this and wants to help, but is unsure of what to do since the branch the cat is on is too small to support his weight (and the cat is too terrified to climb down on its own). Thinking quickly, he goes and grabs some rope and lassos the end of the branch pulling it down hoping the cat will simply climb down. However, the cat swats at him causing him to let of the rope in surprise and send the kitty flying. A few weeks later, the elderly woman was in the grocery store when she notices a younger lady buying some cat food. When she asked the younger lady if she had a cat, the lady replies, "Yes, we named him Angel, since he fell from the sky and into my lap like a miracle."

Literature

Used by Special Forces soldiers in John Scalzi's The Ghost Brigades to escape an enclosure guarded by automatic turrets that lacked the ability to aim up.
In Little Men, the boys are playing by climbing up young trees until they bend all the way down and drop the boys back onto the ground. Jack picks too big a tree, leaving him dangling several feet in the air. Dan climbs up after him, and their combined weight lowers Jack safely on the ground, but as soon as he lets go, Dan goes flying.
- This game is also referenced in Robert Frost's poem "Birches".
In one of the Little House on the Prairie Rose sequels, Swiney shows Rose how to grab the top of a young sapling and jump up and down until their momentum and the springiness of the tree carries them all the way over the top. Rose accidentally lets go at the apex of her flight. Wheeee!

Live Action TV

MythBusters tested this trope in one episode, and found that the right sort of evergreen tree, stripped of its limbs and bent over to the breaking point, could toss Buster the test dummy about hundred yards. However, it still failed to clear a "castle wall" and this trope was deemed "busted".
A slightly more realistic version is used in the MacGyver episode "The Road Not Taken". When he needs to throw off some pursuers in the jungle, MacGyver builds a tree-based catapult to throw stones, and puts a light-based fuse on it. First he pulls four flimsy trees together and bends them down to the ground. He routes them under a solid tree branch and ties them together with a thin vine. Then he sets down his friend's rosary to refract sunlight onto the thin vine, creating a fuse. He stakes the thin vine into the ground with a good knot on a pointed stick, and attaches the pockets from his jacket (filled with rocks) to the tree trunks.

Myths & Religion

This is Older Than Feudalism: In Greek myth, there was a bandit who killed travellers with this trick. Theseus hoisted him by his own petard by letting go early.

Video Games

In Tak and the Power of Juju, orangutans are forever flinging sheep and Tak with banana trees while they try to get the fruit. At the very end of the game, the Juju Flora, who has been transformed into an orangutan by Tlaloc, uses this trick to fling Tlaloc into his own pool of Bad Juju, breaking his curse and turning him into a sheep.
In Sonic Unleashed, the Werehog uses this method to get around various stages by bending the trees or lampposts himself, then relaxing his grip.
In The Secret of Monkey Island, you can make Guybrush fall off a cliff to his presumed doom, and a parody of Sierra's Have a Nice Death screens pops up, only to disappear when Guybrush flies up out of the jungle and back onto the cliff. His only explanation: "Rubber tree."

Web Comics

In Order of the Stick, Belkar does this to Durkon after the dwarf had already climbed onto a palm tree (to fight it, of course). As a variant, the tree doesn't send him flying, but instead unbends even further and smacks the dwarf into the ground on the other side. And then his own hammer falls on him.

Western Animation

Very common in Hanna-Barbera cartoons, especially Wacky Races.
Happens a lot in the Pink Panther cartoons and its spin-offs (The Inspector, The Ant and the Aardvark, Crazy Legs Crane), which are set in such environments more often than in Looney Tunes.
A season one episode of The Simpsons had Homer attempt to make a tree snare and end up launching a bunny rabbit over a mile into the horizon.
- And in a halloween episode "Survival of the Fattest", he did it with a full size tree and essentially the entire supporting cast.

Moe: Oy, ya fatass!

Done in Kim Possible a few times.
This has happened to Wile E. Coyote a time or two with trees or cacti. In one case, rather than launching Wile E himself, a cactus launched a misaimed explosive back at the unfortunate coyote.
In the Veggie Tales episode "Madame Blueberry", the main character's treehouse is eventually flung off its tree in this manner.
The Elefun And Friends short, "A Tangled Tale" features a panda trying to do this with bamboo to get over a river.
1960's Filmation The Atom episode "The House of Doom". The Atom uses one of these to fling himself after an escaping henchman's car.
A Foghorn Leghorn cartoon played with this one, with Henery the Chicken Hawk setting up the "snare" trap, and putting corn in it. Foghorn mocks him roundly, about how a smart chicken would just pick up the corn, and then ends it with "A smart chicken would jump over it, like so!" and demonstrates. Too bad he hadn't thought of Henery building a pitfall trap right next to the snare.
George and Junior try this in the MGM short The Henpecked Hobos. Junior cuts the rope too early, while George is trying to explain the plan...
Near the beginning of Goliath II, the titular elephant accidentally gets knocked into a puddle while attempting to knock over a flower.
The Herculoids episode "The Raiders". In a light-hearted moment Igoo uses his great strength to turn a tree into a catapult to throw Dorno into a really high dive into the water.
In My Little Pony Friendship Is Magic, Applejack uses a handy tree branch to catch up with Rainbow Dash in a race in the episode "Fall Weather Friends".
- In "Family Appreciation Day", the Cutie Mark Crusaders get launched by a tree while trying to pick Zap Apples.
Happens in the Jimmy Two-Shoes episode "Lucius Lost". Trapped on a Far Side Island, Jimmy's plan is to use this trope to catapult him and Lucius to safety. When Lucius cuts the rope prematurely (He'd rather leave Jimmy on the island), the tree doesn't move. Once he gets off and complains, then the tree whips, slamming him into the sand.
Francis X. Bushlad uses one in one of his plans to catch Taz in Taz-Mania. It fails as spectacularly as all of his other plans to catch Taz.

Real Life

Monkeys using trees to clear the walls of a research facility in Japan, via a Cracked article.