Spring, Summer, Fall, Winter... and Spring

The film is divided into five segments (the five seasons of the title), each segment depicting a different stage in the life of a Buddhist monk (each segment is roughly ten to twenty years apart, and is physically in the middle of its titular season).Spring
We are introduced the of the very young Buddhist apprentice with his master on a small floating monastery, drifting on a lake in the serene forested mountains of Korea. The apprentice and his master live a basic life of prayer and meditation, using an old rowboat to reach the bank of the lake where they regularly go walking, for exercise and to collect herbs. One day, in a creek amongst the rocky hills, the apprentice torments a fish by tying a small stone to it with string and laughing as it struggles to swim. Shortly after, he does the same to a frog and a snake; his master quietly observes on all three occasions, and that night ties a large, smooth rock to the apprentice as he sleeps. In the morning, he tells his apprentice that he cannot take off the rock until he unties the creatures he tormented - adding that if any of them have died, he will "carry the stone in his heart forever". The boy struggles with the load on his back through the forest, and finds the fish, lying dead on the bottom of the creek, finds the frog still alive and struggling where he left it, and finds the snake in a pool of blood, presumably attacked and killed by another animal, unable to get away. The master watches as the boy begins to cry heavily upon seeing what he has done to the snake.Summer
The apprentice (now in his teenage years) encounters a mother and daughter (dressed in modern clothes, indicating that the film takes place in modern times) walking along the forest path, looking for the lake monastery. The apprentice silently greets them and rows them across the lake to the monastery, where it is revealed that the daughter has an unspecified illness (she displays symptoms of a fever) and has been brought to the Buddhist master by her mother, hoping that she will be healed. The master agrees to take in the teenage girl for a time, and the mother leaves. Over the next few days, the apprentice finds himself sexually attracted to the girl, but is too shy to say anything; however, when he finds her sleeping in front of the Buddha statue, he is unable to resist groping her chest. She wakes up and slaps him, and in a guilty panic the apprentice begins to pray incessantly, something his master notes as strange. The girl seems to forgive him however; eventually, the two wander off into the forest alone and have sex. They repeat the act over the next few nights, hiding their relationship from the master, until he discovers them asleep and naked, drifting around the lake in the rowboat. He wakes them up by pulling the plug out of the boat. Rather than expressing anger or disappointment, he merely warns his apprentice that "lust leads to desire for possesion, and possesion leads to murder", but does tell him that the girl will have to leave. The apprentice reacts emotionally to this, and in the middle of the night runs away from the monastery in pursuit of the girl, taking the monastery's Buddha statue with him.Fall
Many years later, in "Fall" (or "Autumn"), the aging master returns from a supply run to the local village, and by chance glimpses a warrant for the arrest of his former apprentice, wanted for the murder of his wife. Foreseeing the apprentice's return, he modifies the teenage monk garments by hand, and soon afterward the adult apprentice appears in the spiritual door at the lake's edge, still full of anger and carrying the bloodstained knife with which he stabbed his wife. Unwilling to go on, he seals his eyes, mouth and nose in a suicide ritual and sits in front of the newly returned Buddha statue, waiting for death; the master discovers him, and beats him ruthlessly, professing that while he may have killed his wife, he will not kill himself so easily. He ties his bloodied apprentice to the ceiling and sets a candle to slowly burn through the rope, then begins painting "Heart Sutra" on one side of the monastery deck, by dipping his cat's tail into a tin of black paint. The apprentice eventually falls, and beginning his repentance, cuts his hair off and starts carving the Chinese characters out of the wood. As he carves and the master paints, two detectives arrive at the monastery and try to arrest the apprentice, but the master asks them to let him finish his task. The apprentice continues without stopping, and collapses into sleep immediately upon finishing. Seemingly influenced by the soothing presence of the master, the detectives help the old monk paint his apprentice's carvings in orange, green, blue and purple. The apprentice finally wakes up, and is taken away by the detectives. After they leave, the master, knowing he is at his end, builds a pyre in the rowboat. He seals his ears, eyes, nose and mouth with paper in the same suicide ritual and meditates as he is suffocated and burned to death. One can see the tracks of his tears in the paper seals as flames engulf him.Winter
The middle-aged apprentice returns to the frozen lake and to his former home, which has been drifting uninhabited for years. He finds his master's clothes, laid out just before his death, and digs his master's remains out of the frozen rowboat, setting them to rest in the Buddha statue under a waterfall. He finds a book of choreographic meditative stances, and begins to train and exercise relentlessly in the freezing weather. Eventually, a woman comes to the monastery with her baby son and a shawl wrapped around her face. She seeks to leave her son with the monk and flee, but as she tries to leave in the middle of the night, she stumbles into a hole in the ice the monk dug earlier and drowns. Finding her body the next day causes him to tie the monastery's large, circular stone to his body and climb to the summit of the tallest surrounding mountain holding another statue, which he places there....and Spring
Finally, returning to "Spring", the cycle is completed: the new master lives in the monastery with the abandoned baby, now his apprentice. The boy is shown to torment a tortoise and, wandering into the rocky hills, echoes his predecessor, forcing stones into the mouths of a fish, frog and snake.

Question: What building are the mother and daughter the apprentice encounters looking for?
Answer: The lake monastery